This document provides a proof of König's matching theorem. It begins by defining key terms like bipartite graphs, vertex cover, matching and maximum matching. It then states König's theorem - that for any bipartite graph G, the maximum cardinality of a matching n(G) equals the minimum cardinality of a vertex cover t(G). The proof considers G to be a minimal counterexample and examines different cases like if G is disconnected, a path or cycle. It ultimately arrives at a contradiction, showing that the assumption n(G)<t(G) is false, proving König's theorem that n(G)=t(G) for bipartite graphs.

1. A Short Proof of König's
Matching Theorem
Romeo Rizzi
Journal of Graph Theory
報告者: 陳政謙

2. Bipartite graphs

3. Vertex cover

4. Minimum vertex cover

5. Matching

6. Maximum matching

7. The relationship between minimum
vertex cover and maximum matching
• For any graph G = (V, E), n(G) ≤ t(G), where n(G) is the
maximum cardinality of a matching of G and t(G) is the
minimum cardinality of a vertex cover of G.

8. König's theorem
• Let G be a bipartite graph. Then n(G) = t(G), where n(G) is the
maximum cardinality of a matching of G and t(G) is the
minimum cardinality of a vertex cover of G.

9. The proof of König's theorem
• Let G be a minimal counterexample.
…
…
…
n(G) < t(G)
If there is a graph H smaller than G, then n(H) = t(H).

10. The proof of König's theorem
• Then G is connected, is not a circuit, nor a path.
If G is not connected
G
G1 G2
… Gn
There exists at least one component Gi which n(Gi) < t(Gi), where 1 ≤ i ≤ n.
Then G is not a minimal counterexample.

11. The proof of König's theorem
• Then G is connected, is not a circuit, nor a path.
If G is an odd path
…
If G is an even path
…
n(G) = t(G) =
𝑛 −1
2
n(G) = t(G) =
𝑛
2

12. The proof of König's theorem
• Then G is connected, is not a circuit, nor a path.
Bipartite graphs cannot be odd cycle, so this case can be ignored.
If G is an even cycle
…
n(G) = t(G)
…
…
…

13. The proof of König's theorem
• So, G has a node of degree at least 3. Let u be such a node
and v one of its neighbors.
…
…
…
u v
We consider two cases: n(G - v) < n(G) and n(G - v) = n(G)

14. The proof of König's theorem
• Case 1: n(G - v) < n(G)
By minimality, Gv has a cover W’ with |W’| = n(G - v) < n(G).
…
…
…
u v
W’ ∪ {v} is a cover of G with cardinality n(G) at most.
Hence, t(G) ≤ n(G). The contradiction occurs.

15. The proof of König's theorem
• Case 2: n(G - v) = n(G)
There exists a maximum matching M of G having no edge incident at v.…
…
…
u v
f
Let W’ be a cover of Gf with |W’| = n(G - v) = n(G).
Hence, n(G) = t(G). The contradiction occurs.

16. The proof of König's theorem
• By the proof, the contradiction occurs in both
case 1 and case 2.
• So the assumption n(G) < t(G) is not right.
• We conclude that n(G) = t(G) holds in bipartite
graphs.