Minors in Expanding Graphs

Abstract

1 corresponding vertices u i , u j of Γ are connected by an edge. A graph G = (V, E) is (t, α)-expanding if every subset X ⊂ V of size |X| ≤ α|V |/t has at least t|X| external neighbors in G. A graphfor every subset X ⊆ V , where e(X) stands for the number of edges spanned by X in G. Informally, this definition indicates that the edge distribution of G is similar to that of the random graph G |V |,p , where the degree of similarity is controlled by parameter β.Here are the main results of this paper.Theorem 1 Let 0 < α < 1 be a constant. Let G be a (t, α)-expanding graph of order n, and let t ≥ 10. Then G contains a minor with average degree at least Theorem 2 Let G be a (p, β)-jumbled graph of order n such that β = o(np). Then G contains a minor with average degree cn √ p, for an absolute constant c > 0.This statement is an extension of results of A. Thomason [39,40], who studied the case of constant p. It can be also used to derive some of the results of Drier and Linial [12].Theorem 3 Let 2 ≤ s ≤ s ′ be integers. Let G be a K s,s ′ -free graph with average degree r. Then G contains a minor with average degree cr Theorem 4 Let k ≥ 2 and let G be a C 2k -free graph with average degree r. Then G contains a minor with average degree cr This theorem generalizes results of Thomassen [37], Diestel and Rompel [11], and Kühn and Osthus [22], who proved similar statements under the (much more restrictive) assumption that G has girth at least 2k + 1.All of the above results are, up to a constant factor, asymptotically tight (Theorems 1, 2), or are allegedly tight (Theorems 3, 4), where in the latter case the tightness hinges upon widely accepted conjectures from Extremal Graph Theory about the asymptotic behavior of the Turán numbers of K s,s ′ and of C 2k .2

Minors in random regular graphs

“…An answer to the problem would indicate whether expansion alone is sufficient when trying to force a complete minor of the largest possible order in a sparse graph, or whether other parameters are also relevant. Krivelevich and Sudakov [13] showed that we do have ccl(G) ≥ c n/ log n. (They also considered the case when t is not bounded but grows with n.) As observed in [13], this bound can also be deduced from a result of Plotkin, Rao, and Smith [19] on separators in graphs without a large complete minor. Kleinberg and Rubinfeld [12] also considered the same problem but with a weaker definition of expansion.…”

“…For this, they determined the likely value of ccl(G n,p ) for the random graph G n,p with constant edge probability p and compared this with known results on χ(G n,p ). Krivelevich and Sudakov [13] investigated ccl(G) for expanding graphs G and derived the order of magnitude of ccl(G n,p ) from their results when p = n −c , with c < 1. In [9], we extended these results to any p with pn ≥ c for some constant c > 1, which answered a question from [13].…”

“…Krivelevich and Sudakov [13] investigated ccl(G) for expanding graphs G and derived the order of magnitude of ccl(G n,p ) from their results when p = n −c , with c < 1. In [9], we extended these results to any p with pn ≥ c for some constant c > 1, which answered a question from [13]. In particular, we showed that if pn = c for some fixed c > 1 then a.a.s.…”

A randomized embedding algorithm for trees

“…Otherwise we would get a 4-cycle by counting the number of edges between S and its boundary N (S)\S. This simple observation appears to be a powerful tool in attacking various extremal problems and was used in [26] and [21] to resolve several conjectures about cycle lengths and clique-minors in H-free graphs.…”

“…Indeed, it is not hard to show (see, e.g., [21]) that if G is a K s,t -free graph (s ≥ t) with minimum degree d, then all subsets of G of size at most d 1/(t−1) expand by a factor of Θ(d/s). Therefore, K s,t -free graphs with minimum degree d contain all trees of order Ω d 1+1/(t−1) and maximum degree O(d/s).…”