This document summarizes a presentation on boundary properties of factorial graph classes. It begins with definitions of hereditary graph classes and factorial growth rates. Classes with index 1 can have constant, polynomial, exponential, or factorial growth of log2|Xn|. Factorial classes have log2|Xn| = Θ(n log n). Minimal superfactorial classes are then discussed, including the class of chordal bipartite graphs and a class defined by forbidden induced subgraphs. Finally, a sequence of superfactorial bipartite graph classes is presented.

1. Boundary properties of factorial classes of graphs
Victor Zamaraev
Laboratory of Algorithms and Technologies for Networks Analysis (LATNA),
Higher School of Economics
Joint work with
Vadim Lozin, University of Warwick
Workshop on Extremal Graph Theory
6 June 2014

2. Boundary properties of factorial classes of graphs
Introduction
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3. Boundary properties of factorial classes of graphs
Introduction
All considered graphs are simple (undirected, without loops and
without multiple edges).
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4. Boundary properties of factorial classes of graphs
Introduction
All considered graphs are simple (undirected, without loops and
without multiple edges).
Graphs are labeled by natural numbers 1, . . . , n
6
4 5
1
2
3
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5. Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorphism.
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6. Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorphism.
Definition
A class of graphs is hereditary if it is closed under taking induced
subgraphs.
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7. Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorphism.
Definition
A class of graphs is hereditary if it is closed under taking induced
subgraphs.
Exapmle
Let X be a hereditary class and W4 ∈ X. Then C4 ∈ X.
1
2
3 4
5 1
2 3
4
W4 C4 4 / 28

8. Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
forbidden induced subgraphs.
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9. Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
forbidden induced subgraphs.
Let M be a set of graphs. Then Free(M) denotes the set of all
graphs not containing induced subgraphs isomorphic to graphs from
M.
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10. Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
forbidden induced subgraphs.
Let M be a set of graphs. Then Free(M) denotes the set of all
graphs not containing induced subgraphs isomorphic to graphs from
M.
Statement
Class X is hereditary if and only if there exists M such that
X = Free(M).
We say that graphs in X are M-free.
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11. Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
forbidden induced subgraphs.
Let M be a set of graphs. Then Free(M) denotes the set of all
graphs not containing induced subgraphs isomorphic to graphs from
M.
Statement
Class X is hereditary if and only if there exists M such that
X = Free(M).
We say that graphs in X are M-free.
Example
For the class of bipartite graphs M is {C3, C5, C7, . . . }, i.e.
B = Free(C3, C5, C7, . . . ). 5 / 28

12. Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from X.
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13. Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from X.
Example
Let P be the class of all graph.
|Pn| = 2(n
2) = 2n(n−1)/2
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14. Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from X.
Example
Let P be the class of all graph.
|Pn| = 2(n
2) = 2n(n−1)/2
log2 |Pn| = Θ(n2)
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15. Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994)
For every infinite hereditary class X, which is not the class of all
graphs:
log2 |Xn| = 1 −
1
c(X)
n2
2
+ o(n2
), (1)
where c(X) ∈ N is the index of class X.
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16. Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994)
For every infinite hereditary class X, which is not the class of all
graphs:
log2 |Xn| = 1 −
1
c(X)
n2
2
+ o(n2
), (1)
where c(X) ∈ N is the index of class X.
(i) For c(X) > 1, log2 |Xn| = Θ(n2)
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17. Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994)
For every infinite hereditary class X, which is not the class of all
graphs:
log2 |Xn| = 1 −
1
c(X)
n2
2
+ o(n2
), (1)
where c(X) ∈ N is the index of class X.
(i) For c(X) > 1, log2 |Xn| = Θ(n2)
(ii) For c(X) = 1, log2 |Xn| = o(n2)
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18. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
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19. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
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20. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
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21. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
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22. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
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23. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
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24. Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of a function log2 |Xn|?
Scheinerman E.R., Zito J. (1994)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
There are no intermediate growth rates between first four ranges.
For exmaple, there is no hereditary class X with
log2 |Xn| = Θ(
√
n).
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25. Boundary properties of factorial classes of graphs
Introduction
Constant
Polynomial
Exponential
Factorial layer
Classes with index 1
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26. Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
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27. Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomial class: E1 – graphs with at most one edge
( n
2 + 1).
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28. Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomial class: E1 – graphs with at most one edge
( n
2 + 1).
Exponential class: Co + Co (2n−1).
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29. Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomial class: E1 – graphs with at most one edge
( n
2 + 1).
Exponential class: Co + Co (2n−1).
Factorial class: F – forests (nn−2 < |Fn| < n2n).
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30. Boundary properties of factorial classes of graphs
Introduction
Alekseev V.E. (1997)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
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31. Boundary properties of factorial classes of graphs
Introduction
Alekseev V.E. (1997)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
1 Structural characterizations were obtained for the first three
layers.
2 In every of the four layers all minimal classes were found.
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32. Boundary properties of factorial classes of graphs
Introduction
Constant
Polynomial
Exponential
Factorial layer
Classes with index 1
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33. Boundary properties of factorial classes of graphs
Introduction
Balogh J., Bollob´as B., Weinreich D. (2000)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
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34. Boundary properties of factorial classes of graphs
Introduction
Balogh J., Bollob´as B., Weinreich D. (2000)
Constant classes: log2 |Xn| = Θ(1).
Polynomial classes: log2 |Xn| = Θ(log n).
Exponential classes: log2 |Xn| = Θ(n).
Factorial classes: log2 |Xn| = Θ(n log n).
All other classes are superfactorial.
In addition
1 Characterized lower part of the factorial layer, i.e. classes with
|Xn| < n(1+o(1))n.
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35. Boundary properties of factorial classes of graphs
Introduction
Examples of factorial classes:
forests
planar graphs
line graphs
cographs
permutation graphs
threshold graphs
graphs of bounded vertex degree
graphs of bounded clique-width
et al.
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36. Boundary properties of factorial classes of graphs
Introduction
Problem
Characterize factorial layer.
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37. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
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38. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Constant
Polynomial
Exponential
Factorial
Classes with index 1

39. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Constant
Polynomial
Exponential
Factorial
Classes with index 1
? ? ?
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40. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn| = Θ(n log2
n)
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41. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn| = Θ(n log2
n)
CB = Free(C3, C5, C6, C7, . . .)
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42. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn| = Θ(n log2
n)
CB = Free(C3, C5, C6, C7, . . .)
Theorem (Spinrad J. P., 1995)
log2 |CBn| = Θ(n log2
n)
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43. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn| = Θ(n log2
n)
CB = Free(C3, C5, C6, C7, . . .)
Theorem (Spinrad J. P., 1995)
log2 |CBn| = Θ(n log2
n)
Question
Is the class of chordal bipartite graphs a minimal superfactorial?
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44. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Theorem (Dabrowski K., Lozin V.V., Zamaraev V., 2012)
Let X = Free(2C4, 2C4 + e) ∩ CB. Then log2 |Xn| = Θ(n log2
n).
2C4 2C4 + e
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45. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Theorem (Dabrowski K., Lozin V.V., Zamaraev V., 2012)
Let X = Free(2C4, 2C4 + e) ∩ CB. Then log2 |Xn| = Θ(n log2
n).
2C4 2C4 + e
Open question
Is the class Free(2C4, 2C4 + e) ∩ CB a minimal superfactorial?
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46. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs.
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47. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs.
Statement (follows from the results of Lazebnik F., et al., 1995)
For each integer k ≥ 2, the class Bk is superfactorial.
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48. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs.
Statement (follows from the results of Lazebnik F., et al., 1995)
For each integer k ≥ 2, the class Bk is superfactorial.
Infinite sequence of superfactorial classes
B2 ⊃ B3 ⊃ B4 . . . .
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49. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs.
Statement (follows from the results of Lazebnik F., et al., 1995)
For each integer k ≥ 2, the class Bk is superfactorial.
Infinite sequence of superfactorial classes
B2 ⊃ B3 ⊃ B4 . . . .
In this sequence there is no minimal superfactorial class.
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50. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will
say that the sequence converges to a class X if
i≥1
Xi = X.
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51. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will
say that the sequence converges to a class X if
i≥1
Xi = X.
Example
The sequence B2 ⊃ B3 ⊃ B4 . . . converges to the factorial class F
of forests, i.e.
i≥1
Bi = F.
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52. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will
say that the sequence converges to a class X if
i≥1
Xi = X.
Example
The sequence B2 ⊃ B3 ⊃ B4 . . . converges to the factorial class F
of forests, i.e.
i≥1
Bi = F.
Definition
A class X of graphs is a limit class (for the factorial layer) if there
is a sequence of superfactorial classes converging to X.
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53. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class is called boundary (or minimal) if it does not properly
contain any other limit class.
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54. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class is called boundary (or minimal) if it does not properly
contain any other limit class.
Theorem
A finitely defined class is superfactorial if and only if it contains a
boundary class.
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55. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class is called boundary (or minimal) if it does not properly
contain any other limit class.
Theorem
A finitely defined class is superfactorial if and only if it contains a
boundary class.
Theorem
The class of forests is a boundary class.
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56. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There are five more boundary classes, which can be easly obtained
from the class of forests.
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57. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There are five more boundary classes, which can be easly obtained
from the class of forests.
Two of them are:
1 complements of forests;
2 bipartite complements of forests;
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58. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There are five more boundary classes, which can be easly obtained
from the class of forests.
Two of them are:
1 complements of forests;
2 bipartite complements of forests;
1
5
2
6
3
7
4
8
F
1
5
2
6
3
7
4
8
Bipartite complement of F
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59. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There are five more boundary classes, which can be easly obtained
from the class of forests.
Two of them are:
1 complements of forests;
2 bipartite complements of forests;
1
5
2
6
3
7
4
8
F
1
5
2
6
3
7
4
8
Bipartite complement of F
Question
Are there other boundary classes?
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60. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Lozin’s conjecture
Conjecture (Lozin’s conjecture, [Lozin V.V., Mayhill C., Zamaraev V., 2011])
A hereditary graph class X is factorial if and only if at least one of
the following three classes: X ∩ B, X ∩ B и X ∩ S is factorial and
each of these classes is at most factorial.
B – bipartite graphs
B – complements of bipartite graphs
S – split graphs
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61. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .)
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62. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .)
superfactorial superfactorial
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63. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .)
superfactorial superfactorial
i≥1
Bi = F ⊂ B2
i≥1
Bi = F ⊂ CB
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64. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .)
superfactorial superfactorial
i≥1
Bi = F ⊂ B2
i≥1
Bi = F ⊂ CB
Bi ⊆ B2, i ≥ 1 Bi CB, i ≥ 1
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65. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .)
superfactorial superfactorial
i≥1
Bi = F ⊂ B2
i≥1
Bi = F ⊂ CB
Bi ⊆ B2, i ≥ 1 Bi CB, i ≥ 1
Definition
Let X be a superfactorial class and S a boundary subclass
contained in X. We say that S is a proper boundary subclass of X
if there is a sequence of superfactorial subclasses of X converging
to S.
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66. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
Theorem
There are no proper boundary subclasses of chordal bipartite
graphs.
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67. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
Theorem
There are no proper boundary subclasses of chordal bipartite
graphs.
Theorem
The class of forests is the only proper boundary subclass of B2.
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68. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Open problems
Open question
Find a minimal superfactorial class.
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69. Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Open problems
Open question
Find a minimal superfactorial class.
Open question
Is the list of boundary classes we found complete?
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70. Thank you!