Connectivity and minimum degree games
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The document describes two sequential positional games: the connectivity game and the minimum degree game. It begins with an introduction to the games and provides the rules of the connectivity game. It explains that the goal is for the maker to claim a spanning tree and the breaker to create a cut. The critical bias bC is introduced as the threshold where the game outcome changes from maker's win to breaker's win. Probabilistic intuition is provided by analyzing a random version of the game. It is shown that bC is approximately n/ln(n). The proof of the related Erdos-Renyi theorem is outlined.
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Connectivity and minimum degree games
- 1. Prologue The Connectivity Game The Minimum Degree Game Epilogue The Connectivity and Minimum Degree Games Dalya Gartzman Tel Aviv University Based on notes by Michael Krivelevich Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 2. Prologue The Connectivity Game The Minimum Degree Game Epilogue What Are We Playing? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 3. Prologue The Connectivity Game The Minimum Degree Game Epilogue What Are We Playing? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 4. Prologue The Connectivity Game The Minimum Degree Game Epilogue What Are We Playing? CG sequential games with perfect information Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 5. Prologue The Connectivity Game The Minimum Degree Game Epilogue What Are We Playing? CG sequential games with perfect information PG each move consists of claiming a previously-unclaimed position Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 6. Prologue The Connectivity Game The Minimum Degree Game Epilogue What Are We Playing? CG sequential games with perfect information PG each move consists of claiming a previously-unclaimed position MBG the board is a set V , and H ⊂ P(V ) are the winning sets Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 7. Prologue The Connectivity Game The Minimum Degree Game Epilogue Tic-tac-toe as a Maker-Breaker Game Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 8. Prologue The Connectivity Game The Minimum Degree Game Epilogue Tic-tac-toe as a Maker-Breaker Game Maker Vs. Breaker maker=X,breaker=O Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 9. Prologue The Connectivity Game The Minimum Degree Game Epilogue Tic-tac-toe as a Maker-Breaker Game Maker Vs. Breaker maker=X,breaker=O Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 10. Prologue The Connectivity Game The Minimum Degree Game Epilogue Tic-tac-toe as a Maker-Breaker Game Maker Vs. Breaker maker=X,breaker=O Maker Wins notice the a-symmetry! Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 11. Prologue The Connectivity Game The Minimum Degree Game Epilogue History and Motivation Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 12. Prologue The Connectivity Game The Minimum Degree Game Epilogue History and Motivation Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 13. Prologue The Connectivity Game The Minimum Degree Game Epilogue History and Motivation A Mathematician’s Apology ”Nothing I have ever done is of the slightest practical use” Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 14. Prologue The Connectivity Game The Minimum Degree Game Epilogue History and Motivation A Mathematician’s Apology ”Nothing I have ever done is of the slightest practical use” BUT THEN CAME RSA Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 15. Prologue The Connectivity Game The Minimum Degree Game Epilogue History and Motivation A Mathematician’s Apology ”Nothing I have ever done is of the slightest practical use” BUT THEN CAME RSA So instead of asking useless questions about the usefulness of Mathematics, let‘s just enjoy the fact that we may play games during working time :) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 16. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 17. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 18. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 19. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 20. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 21. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 22. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Game ends anyway when we are out of edges Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 23. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Game ends anyway when we are out of edges What is the Critical Bias? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 24. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Game ends anyway when we are out of edges What is the Critical Bias? When b = 1 Maker plays greedily and wins after n − 1 moves Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 25. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Game ends anyway when we are out of edges What is the Critical Bias? When b = 1 Maker plays greedily and wins after n − 1 moves When b = n − 1 Breaker wins on the first round Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 26. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Connectivity Game Rules of the Game Maker starts by claiming one edge Breaker continues by claiming b ≥ 1 edges If Maker claimed a spanning tree - she wins If Breaker created a cut - he wins Game ends anyway when we are out of edges What is the Critical Bias? When b = 1 Maker plays greedily and wins after n − 1 moves When b = n − 1 Breaker wins on the first round What is the critical bias bC ? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 27. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 28. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 29. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game By the end of the game RandomMaker occupies a random graph (n) n2 2 with 1+b ≈ 2b edges. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 30. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game By the end of the game RandomMaker occupies a random graph (n) n2 2 with 1+b ≈ 2b edges. rand What is bC ? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 31. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game By the end of the game RandomMaker occupies a random graph (n) n2 2 with 1+b ≈ 2b edges. rand What is bC ? THM 1.1 (Erd o s − R´nyi) ¨ e Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 32. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game By the end of the game RandomMaker occupies a random graph (n) n2 2 with 1+b ≈ 2b edges. rand What is bC ? THM 1.1 (Erd o s − R´nyi) ¨ e Consider the random graph G (n, M), and set 0 < 1. If M = (1 − ) n ln n then G (n, M) is not connected WHP, 2 while if M = (1 + ) n ln n then G (n, M) is connected WHP. 2 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 33. Prologue The Connectivity Game The Minimum Degree Game Epilogue Probabilistic Intuition The Random (1 : b) Game By the end of the game RandomMaker occupies a random graph (n) n2 2 with 1+b ≈ 2b edges. rand What is bC ? THM 1.1 (Erd o s − R´nyi) ¨ e Consider the random graph G (n, M), and set 0 < 1. If M = (1 − ) n ln n then G (n, M) is not connected WHP, 2 while if M = (1 + ) n ln n then G (n, M) is connected WHP. 2 Corollary 1.2 rand bC ≈ n2 2M ≈ n ln n Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 34. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 35. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Consider G (n, p) with p(n) = Dalya Gartzman The Connectivity and Minimum Degree Games M (n) 2 (≈ G (n, M)) Tel Aviv University
- 36. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Consider G (n, p) with p(n) = M (n) 2 (≈ G (n, M)) When p = (1 + ) ln n n Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 37. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Consider G (n, p) with p(n) = M (n) 2 (≈ G (n, M)) When p = (1 + ) ln n n Remember: G is connected iff there exists no subset S ⊂ V (G ) such that G has no edges between S and SV . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 38. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Consider G (n, p) with p(n) = M (n) 2 (≈ G (n, M)) When p = (1 + ) ln n n Remember: G is connected iff there exists no subset S ⊂ V (G ) such that G has no edges between S and SV . So we get (details ahead): n→∞ E [# of cuts in G (n, p)] − − 0 −→ Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 39. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 Consider G (n, p) with p(n) = M (n) 2 (≈ G (n, M)) When p = (1 + ) ln n n Remember: G is connected iff there exists no subset S ⊂ V (G ) such that G has no edges between S and SV . So we get (details ahead): n→∞ E [# of cuts in G (n, p)] − − 0 −→ From Markov, this is also an upper bound on the probability that G (n, p) has at least one cut, and hence G is asymptotically almost surely connected. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 40. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the connected case) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 41. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the connected case) By linearity of expectation we get that E [# of cuts in G (n, p)] is n/2 i=1 n (1 − p)i(n−i) ≤ i Dalya Gartzman The Connectivity and Minimum Degree Games n/2 i=1 n −pi(n−i) e i Tel Aviv University
- 42. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the connected case) By linearity of expectation we get that E [# of cuts in G (n, p)] is n/2 i=1 √ n (1 − p)i(n−i) ≤ i n ≤ ne −p(n−i) i=1 Dalya Gartzman The Connectivity and Minimum Degree Games n/2 i=1 n −pi(n−i) e i n/2 i 2n e −p + √ √ n(n− n) √ i= n+1 Tel Aviv University
- 43. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the connected case) By linearity of expectation we get that E [# of cuts in G (n, p)] is n/2 i=1 √ n (1 − p)i(n−i) ≤ i n ≤ ne −p(n−i) 2n e −p + √ √ n(n− n) √ i= n+1 n e− ≤ i=1 n −pi(n−i) e i n/2 i i=1 √ n/2 ln n(1+o(1)) i=1 n/2 i e −Ω(ln n + √ n) √ i= n+1 and both terms tend to zero when n tends to infinity, as desired. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 44. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 45. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 46. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Remember: if G has an isolated vertex, then it is surely disconnected. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 47. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Remember: if G has an isolated vertex, then it is surely disconnected. By linearity of expectation we get E [# of isolated vertices in G (n, p)] = n(1 − p)n−1 ≈ ne −pn Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 48. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Remember: if G has an isolated vertex, then it is surely disconnected. By linearity of expectation we get E [# of isolated vertices in G (n, p)] = n(1 − p)n−1 ≈ ne −pn Since p = (1 − ) ln n we get ne −pn = n → ∞. n Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 49. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Remember: if G has an isolated vertex, then it is surely disconnected. By linearity of expectation we get E [# of isolated vertices in G (n, p)] = n(1 − p)n−1 ≈ ne −pn Since p = (1 − ) ln n we get ne −pn = n → ∞. n Apply Chevishev‘s inequality to get (details ahead): Pr # of isolated vertices in G (n, p) − n ≥ n2 Dalya Gartzman The Connectivity and Minimum Degree Games /3 →0 Tel Aviv University
- 50. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (cont.) When p = (1 − ) ln n n Remember: if G has an isolated vertex, then it is surely disconnected. By linearity of expectation we get E [# of isolated vertices in G (n, p)] = n(1 − p)n−1 ≈ ne −pn Since p = (1 − ) ln n we get ne −pn = n → ∞. n Apply Chevishev‘s inequality to get (details ahead): Pr # of isolated vertices in G (n, p) − n ≥ n2 /3 →0 meaning the probability that there are no isolated vertices, tends to zero when n tends to infinity, as desired. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 51. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 52. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 53. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Var(Z ) = E Z 2 − E2 [Z ] = n(1 − p)n−1 + n(n − 1)(1 − p)2n−3 − n2 (1 − p)2n−2 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 54. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Var(Z ) = E Z 2 − E2 [Z ] = n(1 − p)n−1 + n(n − 1)(1 − p)2n−3 − n2 (1 − p)2n−2 = n(1 − p)n−1 1 − (1 − p)n−2 + n2 (1 − p)2n−2 Dalya Gartzman The Connectivity and Minimum Degree Games p 1−p Tel Aviv University
- 55. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Var(Z ) = E Z 2 − E2 [Z ] = n(1 − p)n−1 + n(n − 1)(1 − p)2n−3 − n2 (1 − p)2n−2 = n(1 − p)n−1 1 − (1 − p)n−2 + n2 (1 − p)2n−2 ≈n 1−n −1 Dalya Gartzman The Connectivity and Minimum Degree Games + n2 p 1−p (1− ) ln n n−(1− ) ln n Tel Aviv University
- 56. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Var(Z ) = E Z 2 − E2 [Z ] = n(1 − p)n−1 + n(n − 1)(1 − p)2n−3 − n2 (1 − p)2n−2 = n(1 − p)n−1 1 − (1 − p)n−2 + n2 (1 − p)2n−2 ≈n 1−n −1 + n2 p 1−p (1− ) ln n n−(1− ) ln n From Chebyshev’s inequality we get: 2 Pr |Z − E[Z ]| ≥ n 3 Dalya Gartzman The Connectivity and Minimum Degree Games ≤ Var(Z ) 4 n3 2 2 ≈ n− 3 − n−1+ 3 + n 3 (1− ) ln n n−(1− ) ln n Tel Aviv University
- 57. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.1 (calculations for the disconnected case) Let Z be the number of isolated vertices in G (n, p), and let’s compute its variance: Var(Z ) = E Z 2 − E2 [Z ] = n(1 − p)n−1 + n(n − 1)(1 − p)2n−3 − n2 (1 − p)2n−2 = n(1 − p)n−1 1 − (1 − p)n−2 + n2 (1 − p)2n−2 ≈n 1−n −1 + n2 p 1−p (1− ) ln n n−(1− ) ln n From Chebyshev’s inequality we get: 2 Pr |Z − E[Z ]| ≥ n 3 ≤ Var(Z ) 4 n3 2 2 ≈ n− 3 − n−1+ 3 + n 3 (1− ) ln n n−(1− ) ln n All three terms tend to zero as n tends to infinity, giving the desired result. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 58. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 59. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 60. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Yes. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 61. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Yes. On our next segment we will show that the threshold biases in the random game, and in the clever game, are equal. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 62. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Yes. On our next segment we will show that the threshold biases in the random game, and in the clever game, are equal. Do all graph properties behave in this manner? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 63. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Yes. On our next segment we will show that the threshold biases in the random game, and in the clever game, are equal. Do all graph properties behave in this manner? No. rand I.e. take diameter-2 game, where bg = Dalya Gartzman The Connectivity and Minimum Degree Games n3/2 √ 2 ln n while bg = 1! Tel Aviv University
- 64. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Does this Intuition hold? Yes. On our next segment we will show that the threshold biases in the random game, and in the clever game, are equal. Do all graph properties behave in this manner? No. 3/2 rand √ I.e. take diameter-2 game, where bg = 2n ln n while bg = 1! In fact, one of the major questions you may ask in this regard, is whether this connection appears, and under what circumstances. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 65. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 66. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 67. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 68. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. ASI gives us an estimate on the outcome of the Clever Game by simulating a Random Game. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 69. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. ASI gives us an estimate on the outcome of the Clever Game by simulating a Random Game. Also, the Clever Game ends much faster... Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 70. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. ASI gives us an estimate on the outcome of the Clever Game by simulating a Random Game. Also, the Clever Game ends much faster... API and AI Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 71. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. ASI gives us an estimate on the outcome of the Clever Game by simulating a Random Game. Also, the Clever Game ends much faster... API and AI Using a Monte-Carlo algorithm, a machine can choose its moves wisely, by simulating random games resulting from each possible move, and act according to the one that gives the best winning chances. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 72. Prologue The Connectivity Game The Minimum Degree Game Epilogue Asymptotic Probabilistic Intuition So should I play randomly? No. Play clever. ASI gives us an estimate on the outcome of the Clever Game by simulating a Random Game. Also, the Clever Game ends much faster... API and AI Using a Monte-Carlo algorithm, a machine can choose its moves wisely, by simulating random games resulting from each possible move, and act according to the one that gives the best winning chances. This technique allowed an unprecedented breakthrough in AI for games. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 73. Prologue The Connectivity Game The Minimum Degree Game Epilogue Cutting to the Chase Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 74. Prologue The Connectivity Game The Minimum Degree Game Epilogue Cutting to the Chase THM 1.3 (Gebauer − Szab´ ) o For every > 0 there is an n0 = n0 ( ) such that for every n ≥ n0 , Maker can build a spanning tree of Kn while playing against a n Breaker who plays with bias (1 − ) ln n . n In particular bC = (1 + o(1)) ln n . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 75. Prologue The Connectivity Game The Minimum Degree Game Epilogue Cutting to the Chase THM 1.3 (Gebauer − Szab´ ) o For every > 0 there is an n0 = n0 ( ) such that for every n ≥ n0 , Maker can build a spanning tree of Kn while playing against a n Breaker who plays with bias (1 − ) ln n . n In particular bC = (1 + o(1)) ln n . (We only show she lower bound today) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 76. Prologue The Connectivity Game The Minimum Degree Game Epilogue Cutting to the Chase THM 1.3 (Gebauer − Szab´ ) o For every > 0 there is an n0 = n0 ( ) such that for every n ≥ n0 , Maker can build a spanning tree of Kn while playing against a n Breaker who plays with bias (1 − ) ln n . n In particular bC = (1 + o(1)) ln n . (We only show she lower bound today) The Plan Maker will abandon the intuitive approach of building a spanning tree, and adopt a strategy of ”breaking” into every cut. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 77. Prologue The Connectivity Game The Minimum Degree Game Epilogue Cutting to the Chase THM 1.3 (Gebauer − Szab´ ) o For every > 0 there is an n0 = n0 ( ) such that for every n ≥ n0 , Maker can build a spanning tree of Kn while playing against a n Breaker who plays with bias (1 − ) ln n . n In particular bC = (1 + o(1)) ln n . (We only show she lower bound today) The Plan Maker will abandon the intuitive approach of building a spanning tree, and adopt a strategy of ”breaking” into every cut. Succeeding with this will imply connectivity, and maintaining a cycle free graph will imply completing a spanning tree. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 78. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 79. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 80. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . component = a connected component on Maker‘s graph Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 81. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . component = a connected component on Maker‘s graph C (v ) = the connected component of vertex v Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 82. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . component = a connected component on Maker‘s graph C (v ) = the connected component of vertex v The degree of a vertex v (or deg(v )) = the degree of v in Breaker‘s graph Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 83. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . component = a connected component on Maker‘s graph C (v ) = the connected component of vertex v The degree of a vertex v (or deg(v )) = the degree of v in Breaker‘s graph A dangerous component = a component that contains at most 2b vertices Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 84. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Setting n Let b=(1 − ) ln n . component = a connected component on Maker‘s graph C (v ) = the connected component of vertex v The degree of a vertex v (or deg(v )) = the degree of v in Breaker‘s graph A dangerous component = a component that contains at most 2b vertices A non-dangerous component is connected to the rest of the graph by at least 2b(n − 2b) > (n − 1)b edges Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 85. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 86. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 87. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 Dalya Gartzman The Connectivity and Minimum Degree Games if v is in a dangerous component otherwise Tel Aviv University
- 88. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 if v is in a dangerous component otherwise MS Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 89. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 if v is in a dangerous component otherwise MS At first all the vertices are active Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 90. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 if v is in a dangerous component otherwise MS At first all the vertices are active On each round, pick an active vertex vC with maximal danger Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 91. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 if v is in a dangerous component otherwise MS At first all the vertices are active On each round, pick an active vertex vC with maximal danger Connect vC to an arbitrary component C Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 92. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Maker’s Strategy Goal: Maintain an active vertex vC in each component C . Tool: define a danger function dang(v ) = deg(v ) 0 if v is in a dangerous component otherwise MS At first all the vertices are active On each round, pick an active vertex vC with maximal danger Connect vC to an arbitrary component C Deactivate vC and assign vC ∪C = vC Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 93. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 94. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 95. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 96. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, and let (K , V K ) be the cut that breaker occupied. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 97. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, and let (K , V K ) be the cut that breaker occupied. Assume WLOG that |K | ≤ |V K |, Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 98. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, and let (K , V K ) be the cut that breaker occupied. Assume WLOG that |K | ≤ |V K |, and let vg ∈ K be an arbitrary active vertex. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 99. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, and let (K , V K ) be the cut that breaker occupied. Assume WLOG that |K | ≤ |V K |, and let vg ∈ K be an arbitrary active vertex. Notice: |K | ≤ 2b, dang(vg ) ≥ n − 2b. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 100. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be the winning round, and let (K , V K ) be the cut that breaker occupied. Assume WLOG that |K | ≤ |V K |, and let vg ∈ K be an arbitrary active vertex. Notice: |K | ≤ 2b, dang(vg ) ≥ n − 2b. Let Mi , Bi be Maker’s and Breaker’s moves on round i. Let vi be the vertex Maker deactivated on round i, and let Ji = {vi+1 , ..., vg }, J = J0 . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 101. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win (cont.) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 102. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win (cont.) Let dang (Mi ) = v ∈Ji−1 dang(v ) |Ji−1 | Dalya Gartzman The Connectivity and Minimum Degree Games dang (Bi ) = dang(v ) |Ji | v ∈Ji Tel Aviv University
- 103. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win (cont.) Let dang (Mi ) = v ∈Ji−1 dang(v ) |Ji−1 | dang (Bi ) = dang(v ) |Ji | v ∈Ji Notice: dang (Mg ) = dang (vg ) ≥ n − 2b, dang (M1 ) = 0. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 104. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Proof of Maker’s win (cont.) Let dang (Mi ) = v ∈Ji−1 dang(v ) |Ji−1 | dang (Bi ) = dang(v ) |Ji | v ∈Ji Notice: dang (Mg ) = dang (vg ) ≥ n − 2b, dang (M1 ) = 0. Goal: show the average danger cannot change so drastically (from 0 to almost n) in less than n rounds, to get a contradiction. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 105. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Maker’s move Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 106. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Maker’s move Lemma 1.3.1 For every i, 1 ≤ i ≤ g − 1, we get dang (Mi ) ≥ dang (Bi ) . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 107. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Maker’s move Lemma 1.3.1 For every i, 1 ≤ i ≤ g − 1, we get dang (Mi ) ≥ dang (Bi ) . Proof of Lemma 1.3.1: Follows immediately from MS . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 108. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 109. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Lemma 1.3.2 For every i, 1 ≤ i ≤ g − 1, we get dang (Bi ) ≥ dang (Mi+1 ) − b + a(i) − a(i − 1) − 1, |Ji | where a(i) is the number of edges Breaker occupied within Ji during the first i rounds. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 110. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Lemma 1.3.2 For every i, 1 ≤ i ≤ g − 1, we get dang (Bi ) ≥ dang (Mi+1 ) − b + a(i) − a(i − 1) − 1, |Ji | where a(i) is the number of edges Breaker occupied within Ji during the first i rounds. Proof of Lemma 1.3.2: Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 111. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Lemma 1.3.2 For every i, 1 ≤ i ≤ g − 1, we get dang (Bi ) ≥ dang (Mi+1 ) − b + a(i) − a(i − 1) − 1, |Ji | where a(i) is the number of edges Breaker occupied within Ji during the first i rounds. Proof of Lemma 1.3.2: Let edouble be the number of edges Breaker adds to Ji during Bi . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 112. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Lemma 1.3.2 For every i, 1 ≤ i ≤ g − 1, we get dang (Bi ) ≥ dang (Mi+1 ) − b + a(i) − a(i − 1) − 1, |Ji | where a(i) is the number of edges Breaker occupied within Ji during the first i rounds. Proof of Lemma 1.3.2: Let edouble be the number of edges Breaker adds to Ji during Bi . So we get dang (Bi ) ≥ dang (Mi+1 ) − Dalya Gartzman The Connectivity and Minimum Degree Games b+edouble . |Ji | Tel Aviv University
- 113. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Change during Breaker’s move Lemma 1.3.2 For every i, 1 ≤ i ≤ g − 1, we get dang (Bi ) ≥ dang (Mi+1 ) − b + a(i) − a(i − 1) − 1, |Ji | where a(i) is the number of edges Breaker occupied within Ji during the first i rounds. Proof of Lemma 1.3.2: Let edouble be the number of edges Breaker adds to Ji during Bi . So we get dang (Bi ) ≥ dang (Mi+1 ) − b+edouble . |Ji | To finish, notice that the number of edges taken by Breaker until Bi−1 in Ji = a(i) − edouble , but is also ≥ a(i − 1) − |Ji |. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 114. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 115. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 116. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . This, together with the application of Lemma 1.3.2 on the last k rounds of the game, gives (details ahead) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 117. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . This, together with the application of Lemma 1.3.2 on the last k rounds of the game, gives (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 3) − Dalya Gartzman The Connectivity and Minimum Degree Games n , ln n Tel Aviv University
- 118. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . This, together with the application of Lemma 1.3.2 on the last k rounds of the game, gives (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 3) − n , ln n whereas if the game ended after less than k rounds, we get (details ahead) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 119. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . This, together with the application of Lemma 1.3.2 on the last k rounds of the game, gives (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 3) − n , ln n whereas if the game ended after less than k rounds, we get (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + 3) − Dalya Gartzman The Connectivity and Minimum Degree Games n . ln n Tel Aviv University
- 120. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - Almost the end n Let k = ln n , and notice that during the first g − k − 1 rounds of 2b the game we surely get dang (Bi ) ≥ dang (Mi+1 ) − |Ji | . This, together with the application of Lemma 1.3.2 on the last k rounds of the game, gives (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 3) − n , ln n whereas if the game ended after less than k rounds, we get (details ahead) 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + 3) − n . ln n n On both cases, b must be at least (1 + o(1)) ln n to satisfy these inequalities, contradicting our assumption, and finishing the proof. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 121. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 122. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − Dalya Gartzman The Connectivity and Minimum Degree Games 2b g −1 ≥ ··· Tel Aviv University
- 123. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | 2b − k+1 − · · · − − ··· − 2b g −1 ≥ ··· b+a(g −k)−a(g −k−1) |Jg −k | −k 2b g −1 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 124. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | 2b − k+1 − · · · − ♠ ≥ dang (Mg ) − b 1 − ··· − 2b g −1 ≥ ··· b+a(g −k)−a(g −k−1) |Jg −k | −k 2b k+1 2b g −1 2b g −1 − b 2 Dalya Gartzman The Connectivity and Minimum Degree Games −···− b k − a(g −1) 1 −k − −···− Tel Aviv University
- 125. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | 2b − k+1 − · · · − ♠ ≥ dang (Mg ) − b 1 − ··· − 2b g −1 ≥ ··· b+a(g −k)−a(g −k−1) |Jg −k | −k 2b k+1 2b g −1 2b g −1 − b 2 −···− b k − a(g −1) 1 −k − −···− ≥ (n − 2b) − b(1 + ln k) − k − 2b(ln g − ln k) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 126. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | 2b − k+1 − · · · − ♠ ≥ dang (Mg ) − b 1 − ··· − 2b g −1 ≥ ··· b+a(g −k)−a(g −k−1) |Jg −k | −k 2b k+1 2b g −1 2b g −1 − b 2 −···− b k − a(g −1) 1 −k − −···− ≥ (n − 2b) − b(1 + ln k) − k − 2b(ln g − ln k) ≥ n − b(ln n + ln ln n + 3) − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n Tel Aviv University
- 127. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the long game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | 2b − k+1 − · · · − ♠ ≥ dang (Mg ) − b 1 − ··· − 2b g −1 ≥ ··· b+a(g −k)−a(g −k−1) |Jg −k | −k 2b k+1 2b g −1 2b g −1 − b 2 −···− b k − a(g −1) 1 −k − −···− ≥ (n − 2b) − b(1 + ln k) − k − 2b(ln g − ln k) n ≥ n − b(ln n + ln ln n + 3) − ln n Where in ♠ we used the fact that a(g − ) for all , and that a(g − 1) = 0 since Jg −1 vertex, and hence no edges. Dalya Gartzman The Connectivity and Minimum Degree Games 1 − 1 |Jg − +1 | |Jg − | contains only one ≥0 Tel Aviv University
- 128. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 129. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − Dalya Gartzman The Connectivity and Minimum Degree Games b+a(1)−a(0) |J1 | − 1 ≥ ··· Tel Aviv University
- 130. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | Dalya Gartzman The Connectivity and Minimum Degree Games − ··· − b+a(1)−a(0) |J1 | b+a(1)−a(0) |J1 | − 1 ≥ ··· −g Tel Aviv University
- 131. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | b 1 − b 2 Dalya Gartzman The Connectivity and Minimum Degree Games − ··· − − ··· − b g −1 − b+a(1)−a(0) |J1 | b+a(1)−a(0) |J1 | a(g −1) 1 − 1 ≥ ··· −g −g Tel Aviv University
- 132. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | b 1 − b 2 − ··· − − ··· − b g −1 − b+a(1)−a(0) |J1 | b+a(1)−a(0) |J1 | a(g −1) 1 − 1 ≥ ··· −g −g ≥ (n − 2b) − b(1 + ln g ) − g Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 133. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | b 1 − b 2 − ··· − − ··· − b g −1 − b+a(1)−a(0) |J1 | b+a(1)−a(0) |J1 | a(g −1) 1 − 1 ≥ ··· −g −g ≥ (n − 2b) − b(1 + ln g ) − g ≥ n − b(ln n + 3) − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n Tel Aviv University
- 134. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 1.3 - calculations for the short game version 0 = dang (M1 ) ≥ dang (B1 ) ≥ dang (M2 ) − ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(g −1)−a(g −2) |Jg −1 | b 1 − b 2 − ··· − − ··· − b g −1 − b+a(1)−a(0) |J1 | b+a(1)−a(0) |J1 | a(g −1) 1 − 1 ≥ ··· −g −g ≥ (n − 2b) − b(1 + ln g ) − g ≥ n − b(ln n + 3) − n ln n Where in ♣ we used the same argument as before. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 135. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 136. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 137. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. rand What is bDc ? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 138. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. rand What is bDc ? THM 2.1 (Erd o s − R´nyi) ¨ e Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 139. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. rand What is bDc ? THM 2.1 (Erd o s − R´nyi) ¨ e Consider the random graph G (n, p), and set 0 < if p = (1 − ) ln n then δ (G (n, p)) < c WHP, n while if p = (1 + ) ln n then δ (G (n, p)) ≥ c WHP. n Dalya Gartzman The Connectivity and Minimum Degree Games 1. Tel Aviv University
- 140. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. rand What is bDc ? THM 2.1 (Erd o s − R´nyi) ¨ e Consider the random graph G (n, p), and set 0 < if p = (1 − ) ln n then δ (G (n, p)) < c WHP, n while if p = (1 + ) ln n then δ (G (n, p)) ≥ c WHP. n 1. (THM 2.1 can be proven in a similar way as THM 1.1) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 141. Prologue The Connectivity Game The Minimum Degree Game Epilogue Introducing The Minimum Degree Game Goal To win Dc , Maker needs to obtain the edges of a spanning subgraph of Kn of minimum degree c. rand What is bDc ? THM 2.1 (Erd o s − R´nyi) ¨ e Consider the random graph G (n, p), and set 0 < if p = (1 − ) ln n then δ (G (n, p)) < c WHP, n while if p = (1 + ) ln n then δ (G (n, p)) ≥ c WHP. n 1. (THM 2.1 can be proven in a similar way as THM 1.1) We will now show that Dc satisfies API. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 142. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 143. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc THM 2.2 (Gebauer − Szab´ ) o n bDc = (1 + o(1)) ln n . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 144. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc THM 2.2 (Gebauer − Szab´ ) o n bDc = (1 + o(1)) ln n . (Again, we only show the lower bound) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 145. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc THM 2.2 (Gebauer − Szab´ ) o n bDc = (1 + o(1)) ln n . (Again, we only show the lower bound) n Set b = (1 − ) ln n Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 146. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc THM 2.2 (Gebauer − Szab´ ) o n bDc = (1 + o(1)) ln n . (Again, we only show the lower bound) n Set b = (1 − ) ln n Let dM (v ), dB (v ) be the degrees of vertex v in Maker’s, and Breaker’s graphs, respectively Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 147. Prologue The Connectivity Game The Minimum Degree Game Epilogue Setting the stage for proving bDc THM 2.2 (Gebauer − Szab´ ) o n bDc = (1 + o(1)) ln n . (Again, we only show the lower bound) n Set b = (1 − ) ln n Let dM (v ), dB (v ) be the degrees of vertex v in Maker’s, and Breaker’s graphs, respectively A vertex v is dangerous if dM (v ) < c Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 148. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 149. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 150. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS In round i, pick an arbitrary dangerous vertex vi with the largest danger, and occupy an arbitrary edge incident to vi . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 151. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS In round i, pick an arbitrary dangerous vertex vi with the largest danger, and occupy an arbitrary edge incident to vi . This action is called easing vi . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 152. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS In round i, pick an arbitrary dangerous vertex vi with the largest danger, and occupy an arbitrary edge incident to vi . This action is called easing vi . Maker is going to have a hard time if some vertex v satisfies dM (v ) + Dalya Gartzman The Connectivity and Minimum Degree Games n−1−dB (v )−dM (v ) b+1 <c Tel Aviv University
- 153. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS In round i, pick an arbitrary dangerous vertex vi with the largest danger, and occupy an arbitrary edge incident to vi . This action is called easing vi . Maker is going to have a hard time if some vertex v satisfies dM (v ) + n−1−dB (v )−dM (v ) b+1 <c ⇒ Make damage control and be sure that every v satisfies dB (v ) − b · dM (v ) ≤ n − 1 − c(b + 1) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 154. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 MS In round i, pick an arbitrary dangerous vertex vi with the largest danger, and occupy an arbitrary edge incident to vi . This action is called easing vi . Maker is going to have a hard time if some vertex v satisfies dM (v ) + n−1−dB (v )−dM (v ) b+1 <c ⇒ Make damage control and be sure that every v satisfies dB (v ) − b · dM (v ) ≤ n − 1 − c(b + 1) ⇒ Let the danger function for the game be dang(v ) = dB (v ) − 2b · dM (v ) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 155. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 156. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 157. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 158. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Then by this time he occupied at least n − c edges incident to some vertex vg . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 159. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Then by this time he occupied at least n − c edges incident to some vertex vg . Maker is easing a vertex every turn, hence g − 1 < cn. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 160. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Then by this time he occupied at least n − c edges incident to some vertex vg . Maker is easing a vertex every turn, hence g − 1 < cn. Since dM (vg ) < c at the end fo the game, we get dang (vg ) > n − c − 2bc = (1 − o(1)) n. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 161. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Then by this time he occupied at least n − c edges incident to some vertex vg . Maker is easing a vertex every turn, hence g − 1 < cn. Since dM (vg ) < c at the end fo the game, we get dang (vg ) > n − c − 2bc = (1 − o(1)) n. Let vi be the vertex that Maker eased on her i-th turn, and define the set Ji = {vi+1 , ..., vg }. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 162. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - cont. Assume by contradiction that Breaker has a winning strategy against MS . Let g − 1 be Breaker’s winning round. Then by this time he occupied at least n − c edges incident to some vertex vg . Maker is easing a vertex every turn, hence g − 1 < cn. Since dM (vg ) < c at the end fo the game, we get dang (vg ) > n − c − 2bc = (1 − o(1)) n. Let vi be the vertex that Maker eased on her i-th turn, and define the set Ji = {vi+1 , ..., vg }. Define dang (Mi ) , dang (Bi ) as before, and get: dang (Mg ) = dang(vg ) ≥ (1 − o(1))n , dang (M1 ) = 0, a difference we will now show to be impossible. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 163. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Maker’s move Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 164. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Maker’s move Lemma 2.2.1 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 165. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Maker’s move Lemma 2.2.1 For every i, 1 ≤ i ≤ g − 1, we get dang (Mi ) ≥ dang (Bi ) . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 166. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Maker’s move Lemma 2.2.1 For every i, 1 ≤ i ≤ g − 1, we get dang (Mi ) ≥ dang (Bi ) . If Ji = Ji−1 , then also dang (Mi ) ≥ dang (Bi ) + Dalya Gartzman The Connectivity and Minimum Degree Games 2b . |Ji | Tel Aviv University
- 167. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Maker’s move Lemma 2.2.1 For every i, 1 ≤ i ≤ g − 1, we get dang (Mi ) ≥ dang (Bi ) . If Ji = Ji−1 , then also dang (Mi ) ≥ dang (Bi ) + 2b . |Ji | Proof of Lemma 2.2.1: Follows immediately from MS . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 168. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Breaker’s move Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 169. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Breaker’s move Lemma 2.2.2 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 170. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Breaker’s move Lemma 2.2.2 For every i, 1 ≤ i ≤ g − 1, we get the two lower bounds 2b |Ji | b + a(i) − a(i − 1) dang (Bi ) ≥ dang (Mi+1 ) − −1 |Ji | dang (Bi ) ≥ dang (Mi+1 ) − (1) (2) where a(i) is the number of edges Breaker occupies within Ji during the first i rounds. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 171. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change during Breaker’s move Lemma 2.2.2 For every i, 1 ≤ i ≤ g − 1, we get the two lower bounds 2b |Ji | b + a(i) − a(i − 1) dang (Bi ) ≥ dang (Mi+1 ) − −1 |Ji | dang (Bi ) ≥ dang (Mi+1 ) − (1) (2) where a(i) is the number of edges Breaker occupies within Ji during the first i rounds. Proof of Lemma 2.2.2: Exactly the same arguments as on the Connectivity game. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 172. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change between Maker’s moves Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 173. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change between Maker’s moves Corollary 2.2.3 Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 174. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change between Maker’s moves Corollary 2.2.3 For every i, 1 ≤ i ≤ g − 1, we get the two lower bounds dang (Mi ) ≥ dang (Mi+1 ) , provided Ji = Ji−1 dang (Mi ) ≥ dang (Mi+1 ) − min (3) 2b b + a(i) − a(i − 1) , −1 |Ji | |Ji | (4) where a(i) is the number of edges Breaker occupies within Ji during the first i rounds. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 175. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Change between Maker’s moves Corollary 2.2.3 For every i, 1 ≤ i ≤ g − 1, we get the two lower bounds dang (Mi ) ≥ dang (Mi+1 ) , provided Ji = Ji−1 dang (Mi ) ≥ dang (Mi+1 ) − min (3) 2b b + a(i) − a(i − 1) , −1 |Ji | |Ji | (4) where a(i) is the number of edges Breaker occupies within Ji during the first i rounds. So the interesting situation is when Ji = Ji+1 , then the danger might grow. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 176. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 177. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 178. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Notice: |Jir − | = + 1, |Jir | = |Jg −1 | = 1, |Ji1 −1 | = |J0 | = r + 1. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 179. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Notice: |Jir − | = + 1, |Jir | = |Jg −1 | = 1, |Ji1 −1 | = |J0 | = r + 1. n Let k = ln n . Applying the estimates on Corollary 2.2.3 we get (details ahead): Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 180. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Notice: |Jir − | = + 1, |Jir | = |Jg −1 | = 1, |Ji1 −1 | = |J0 | = r + 1. n Let k = ln n . Applying the estimates on Corollary 2.2.3 we get (details ahead): If r ≥ k 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 2c + 1) − c − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n , Tel Aviv University
- 181. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Notice: |Jir − | = + 1, |Jir | = |Jg −1 | = 1, |Ji1 −1 | = |J0 | = r + 1. n Let k = ln n . Applying the estimates on Corollary 2.2.3 we get (details ahead): If r ≥ k 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 2c + 1) − c − n ln n , and if r < k 0 = dang (M1 ) ≥ · · · ≥ n − b(2c + ln r − +1) − c − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n . Tel Aviv University
- 182. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - Almost the end Let 1 ≤ i1 ≤ ... ≤ ir ≤ g − 1 the indices where Ji = Ji−1 . Notice: |Jir − | = + 1, |Jir | = |Jg −1 | = 1, |Ji1 −1 | = |J0 | = r + 1. n Let k = ln n . Applying the estimates on Corollary 2.2.3 we get (details ahead): If r ≥ k 0 = dang (M1 ) ≥ · · · ≥ n − b(ln n + ln ln n + 2c + 1) − c − n ln n , and if r < k 0 = dang (M1 ) ≥ · · · ≥ n − b(2c + ln r − +1) − c − n ln n . n On both cases, b must be at least (1 + o(1)) ln n to satisfy these inequalities, contradicting our assumption, and finishing the proof. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 183. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 184. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − Dalya Gartzman The Connectivity and Minimum Degree Games 2b |Ji1 | ≥ ··· Tel Aviv University
- 185. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − ≥ dang (Mg ) − − 2b Jir −k b+a(ir )−a(ir −1) |Jir | − ··· − −···− 2b |Ji1 | ≥ ··· b+a(ir −k+1 )−a(ir −k+1 −1) Jir −k+1 −k 2b |Ji1 | Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 186. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − ≥ dang (Mg ) − − ♠ 2b Jir −k b+a(ir )−a(ir −1) |Jir | − ··· − ≥ dang (Mg ) − b 1 −···− 2b |Ji1 | ≥ ··· b+a(ir −k+1 )−a(ir −k+1 −1) Jir −k+1 −k 2b |Ji1 | − b 2 Dalya Gartzman The Connectivity and Minimum Degree Games − ··· − b k − a(ir ) 1 −k − 2b k+1 − ··· − 2b g −1 Tel Aviv University
- 187. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − ≥ dang (Mg ) − − ♠ 2b Jir −k b+a(ir )−a(ir −1) |Jir | − ··· − ≥ dang (Mg ) − b 1 −···− 2b |Ji1 | ≥ ··· b+a(ir −k+1 )−a(ir −k+1 −1) Jir −k+1 −k 2b |Ji1 | − b 2 − ··· − b k − a(ir ) 1 −k − 2b k+1 − ··· − 2b g −1 ≥ (n − c − 2bc) − b(1 + ln k) − k − 2b(ln n − ln k) Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 188. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − ≥ dang (Mg ) − − ♠ 2b Jir −k b+a(ir )−a(ir −1) |Jir | − ··· − ≥ dang (Mg ) − b 1 −···− 2b |Ji1 | ≥ ··· b+a(ir −k+1 )−a(ir −k+1 −1) Jir −k+1 −k 2b |Ji1 | − b 2 − ··· − b k − a(ir ) 1 −k − 2b k+1 − ··· − 2b g −1 ≥ (n − c − 2bc) − b(1 + ln k) − k − 2b(ln n − ln k) ≥ n − b(ln n + ln ln n + 2c + 1) − c − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n Tel Aviv University
- 189. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the long game version 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − ≥ dang (Mg ) − − ♠ 2b Jir −k b+a(ir )−a(ir −1) |Jir | − ··· − ≥ dang (Mg ) − b 1 −···− 2b |Ji1 | ≥ ··· b+a(ir −k+1 )−a(ir −k+1 −1) Jir −k+1 −k 2b |Ji1 | − b 2 − ··· − b k − a(ir ) 1 −k − 2b k+1 − ··· − 2b g −1 ≥ (n − c − 2bc) − b(1 + ln k) − k − 2b(ln n − ln k) ≥ n − b(ln n + ln ln n + 2c + 1) − c − n ln n Where in ♠ we used the fact that a(ir − − 1) ≥ a(ir − −1 ) for all (since Jir −j−1 = Jir −j −1 ), and that a(ir ) = 0 since Jg −1 contains only one vertex, and hence no edges. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 190. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 191. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 192. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | ≥ dang (Mg ) − b+a(ir )−a(ir −1) |Jir | Dalya Gartzman The Connectivity and Minimum Degree Games − ··· − b+a(i2 )−a(i1 ) |Ji1 | −r Tel Aviv University
- 193. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(ir )−a(ir −1) |Jir | b 1 − b 2 Dalya Gartzman The Connectivity and Minimum Degree Games − ··· − − ··· − b r − b+a(i2 )−a(i1 ) |Ji1 | a(ir ) 1 −r −r Tel Aviv University
- 194. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(ir )−a(ir −1) |Jir | b 1 − b 2 − ··· − − ··· − b r − b+a(i2 )−a(i1 ) |Ji1 | a(ir ) 1 −r −r ≥ (n − c − 2bc) − b(1 + ln r ) − r Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 195. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(ir )−a(ir −1) |Jir | b 1 − b 2 − ··· − − ··· − b r − b+a(i2 )−a(i1 ) |Ji1 | a(ir ) 1 −r −r ≥ (n − c − 2bc) − b(1 + ln r ) − r ≥ n − b(2c + ln r − +1) − c − Dalya Gartzman The Connectivity and Minimum Degree Games n ln n Tel Aviv University
- 196. Prologue The Connectivity Game The Minimum Degree Game Epilogue Proof of THM 2.2 - calculations for the short game version 2 0 = dang (M1 ) ≥ dang (Mi1 ) ≥ dang (Mi2 ) − b+a(iJ)−a(i1 ) − 1 ≥ · · · | i1 | ≥ dang (Mg ) − ♣ ≥ dang (Mg ) − b+a(ir )−a(ir −1) |Jir | b 1 − b 2 − ··· − − ··· − b r − b+a(i2 )−a(i1 ) |Ji1 | a(ir ) 1 −r −r ≥ (n − c − 2bc) − b(1 + ln r ) − r ≥ n − b(2c + ln r − +1) − c − n ln n Where in ♣ we used the same argument as before. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 197. Prologue The Connectivity Game The Minimum Degree Game Epilogue Where do we go from here? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 198. Prologue The Connectivity Game The Minimum Degree Game Epilogue Where do we go from here? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 199. Prologue The Connectivity Game The Minimum Degree Game Epilogue Where do we go from here? We know the properties of connectivity and minimum degree 1 appear with high correlation in random graphs. Open conjecture: for all n large enough, bC = bD1 . Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 200. Prologue The Connectivity Game The Minimum Degree Game Epilogue Where do we go from here? We know the properties of connectivity and minimum degree 1 appear with high correlation in random graphs. Open conjecture: for all n large enough, bC = bD1 . What can be a strong criteria for a property to satisfy API? Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 201. Prologue The Connectivity Game The Minimum Degree Game Epilogue Where do we go from here? We know the properties of connectivity and minimum degree 1 appear with high correlation in random graphs. Open conjecture: for all n large enough, bC = bD1 . What can be a strong criteria for a property to satisfy API? Next time - Hemiltonicity game. Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University
- 202. Prologue The Connectivity Game The Minimum Degree Game Epilogue Thank you for listening! Questions? dalyagar@mail.tau.ac.il Dalya Gartzman The Connectivity and Minimum Degree Games Tel Aviv University