Forbidden induced subgraphs for toughness

Transcription

1 (with G. Sueiro) Keio University

2 Definition of Toughness ω(g) = # of components in G Definition A graph G is said to be t-tough S V (G): cutset of G, t ω(g S) S τ(g) = max{ t G is t-tough} : toughness of G Problem What kind of forbidden structures ensures large toughness? Easy to answer: S: cutset such that ω(g S) > S /t.

3 Forbidden structures We consider minors or induced subgraphs as forbidden structures. This is because minor closed families of graphs, and hereditary graph properties involve many important classes of graphs.

4 Forbidden minors Proposition (folklore) If G is a 3-connected planar graph, then τ(g) > 1 2. Theorem (Chen, Egawa, Kawarabayashi, Mohar, O 2011) If G is 3-connected and K 3,t -minor-free, then τ(g) > 1 t 1. 1 is sharp for every odd t. t 1 Theorem (95% true, with Ozeki) If t is even, G is 3-connected and K 3,t -minor-free, then τ(g) > 1 t 2.

5 Forbidden minors Theorem (CEKMO 2011) For any t s 3, c = c(s, t) > 0 such that: If G is s-connected and K s,t -minor-free, then τ(g) > c. Note that: G = K s,n is s-connected, and τ(g) 0 as N. So we have to forbid a minor of K s,t (for some t) in s-connected graphs to ensure a graph to be (c > 0)-tough.

6 Forbidden induced subgraphs G, H: graphs H G H is an induced subgraph of G G is H-free H G (H is forbidden in G) F: a family of graphs G is F-free H F, G is H-free (F is forbidden)

7 Example Theorem (Las Vergnas 1975, Sumner 1976) G is a connected graph of even order. If G is K 1,3 -free, then G has a perfect matching. Theorem (Duffus, Gould, Jacobson 1981) G is a connected graph. If G is {K 1,3, N}-free, then G has a hamiltonian path. K 1,3 N

8 Forbidden induced subgraph problem Problem P: a property of graphs Find all the families of connected graphs F such that all large enough F-free connected graphs satisfy P. We may assume that F consists only of connected graphs. 2K 1 -free P 3 -free { 2K 3 -free,,... } -free

9 Relation of forbidden graphs If H 1 H 2, then every H 1 -free graph is H 2 -free. For two families of connected graphs F 1 and F 2, if H 2 F 2, H 1 F 1 such that H 1 H 2, then every F 1 -free graph is F 2 -free. Definition F 1, F 2 : families of connected graphs F 1 F 2 H 2 F 2, H 1 F 1 such that H 1 H 2.

10 Graphs with a hamiltonian path Theorem (Duffus, Gould, Jacobson 1981) G is a connected graph. If G is {K 1,3, N}-free, then G has a hamiltonian path. Theorem (Faudree, Gould 1997) F: a family of connected graphs with F 2 Then the following are equivalent: Every connected F-free graph has a hamiltonian path; F {K 1,3, N}.

11 Graphs with a hamiltonian cycle Theorem (Faudree, Gould 1997) F: a family of connected graphs with F 2 Then the following are equivalent: Every 2-conn. F-free graph of order 10 is hamiltonian; F {K 1,3, P 6 }, {K 1,3, N}, {K 1,3, W } or {K 1,3, Z 3 }. W Z 3

12 Graphs with a perfect matching Theorem (O, Plummer, Saito 2011) F: a family of connected graphs with F 3 Then the following are equivalent: Every large enough connected F-free graph of even order has a perfect matching; F {K 1,3 }, {K 1,l, P 4, Z 1,r } or {K 1,l, Y m, Z1,r } for some l 4, m 3 and r 3. } l K r K r K 1,l P 4 Z 1,r Z 1,r

13 Graphs with a perfect matching characterization Theorem (O, Sueiro 2012) F: a family of connected graphs. Then the following are equivalent: Every large enough connected F-free graph of even order has a perfect matching; F F M (l, m, q) for some l 4, m 1 and q 2, where F M (l, m, q) = {K 1,l, W q, Y m+2, Z 1,q,..., Z m,q }. K q } {{ } m+2 K q K q } {{ } m W q Y m+2 Z 1,q Z m,q

14 Forbidding for toughness Problem For each real number t > 0, characterize the families F of connected graphs such that: every large enough connected F-free graph is t-tough. suggested by Hajo.

15 Main Theorem (0 < t 1 2 ) Theorem 1 Let 0 < t 1, and F a family of connected graphs. Then the 2 following are equivalent: Every large enough connected F-free graph is t-tough; F F A n (l, m, q), where n = 1/t, for some l n + 2, m 1 and q 3. F A n (l, m, q) = {K 1,l, Y n m+2, Z n 1,q,..., Z n m,q} } {{ } m+2 } n K q }n K q } {{ } m } n Y n m+2 Z n 1,q Z n m,q

16 Main Theorem (t > 1 2 ) Theorem 2 Let t > 1, and F a family of connected graphs. Then the 2 following are equivalent: Every large enough connected F-free graph is t-tough; F F B (l, m, q) for some l 3, m 4 and q 3. F B (l, m, q) = {K 1,l, P m, Z 1,q } } l K q K 1,l P m Z 1,q

17 Outline of Proof Suppose that F is the forbidden family for t-toughness. Choose a series of large graphs each of which is not t-tough. F must contain an induced subgraph from each of the graphs. Conversely, suppose that G is a connected F A n (l, m, q)-free (or F B (l, m.q)-free) graph. Assuming that G is not t-tough, we prove that G is bounded, by showing that (G) and diam(g) are bounded.

18 Remark 1: Forbidden family for t > 1 2 The forbidden family for t-tough with t > 1 2 : F B (l, m, q) = {K 1,l, P m, Z 1,q } Large enough F B (l, m, q)-free connected graphs have arbitrary high toughness. Theorem Large enough F B (l, m, q)-free connected graphs has a hamiltonian cycle. Theorem Large enough F B (l, m, q)-free connected graphs has the square of a hamiltonian cycle. Proposition There exists a constant C = C(l, m.q) > 0 such that every F B (l, m, q)-free connected graph G has δ G C.

19 Remark 2: Forbidden family for 1 3 < t 1 2 The forbidden family for t-tough with 1 3 < t 1 2 : F A 2 (l, m, q) = {K 1,l, Y m+2, Z 1,q,..., Z m,q } Theorem (Furuya 2012+) Every large enough F A 2 (l, m, q)-free connected graph has a 2-walk. Conjecture (Furuya) Every large enough F A n (l, m, q)-free connected graph has a n-walk (n 3).

20 Remark 3: Toughness vs perfect matching The forbidden family for t-tough with 1 3 < t 1 2 : F A 2 (l, m, q) = {K 1,l, Y m+2, Z 1,q,..., Z m,q } The forbidden family for perfect matching: F M (l, m, q) = {K 1,l, W q, Y m+2, Z 1,q,..., Z m,q } Theorem F: a family of connected graphs. Then the following are equivalent: Every large enough connected F-free graph G satisfies S V (G), ω(g S) S + 1; F F M (l, m, q) for some l 4, m 1 and q 2.

21 Thank you!