Minors in random regular graphs

Abstract

ABSTRACT:We show that there is a constant c so that for fixed r ≥ 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c √ n vertices as a minor. This confirms a conjecture of Markström (Ars Combinatoria 70 (2004) 289-295). Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G n,p during the phase transition (i.e., when pn → 1).

Complete Minors in Graphs Without Sparse Cuts

“…For every ε ą 0 there exist β ą 0 and n 0 P N such that the following holds. Let G be a d-regular graph with n ě n 0 vertices, for some d ě 3, and let λ 2 be the second largest eigenvalue of the adjacency matrix of G. If λ 2 ă p1{2´εqd, then cclpGq ě β a nd{ log d. This extends a result by Fountoulakis, Kühn, and Osthus [14] who showed the same statement for constant d ě 3 (whereas we allow d to be a function of n). Optimality of Corollary 1.4 can be derived as follows: Calculations in [13] show that with probability at least 1´expp´Cn logpnpqq we have cclpGpn, pqq " Op a n 2 p{ logpnpqq, for a constant C of our choice (having an impact on the hidden constant in Op¨q).…”

On the Non-Planarity of a Random Subgraph

“…For the case of binomial random graphs G n,p , Fountoulakis, Kühn and Osthus showed [3] that for any c > 1, the random graph G n,p with p = c/n has w.h.p. a complete minor of order √ n. (See also [5] for results for other values of p = p(n), and [4] for results on random regular graphs and for G n,p in the slightly supercritical regime. )…”

Minors in graphs of large θr-girth

“…This result has been proven by Kostochka in [20] and Thomason in [33] and a precise estimation of the constant c has been given by Thomason in [34]. For recent results related to conditions that force a clique minor see [13,15,19,22,23].The girth of a graph G is the minimum length of a cycle in G. Interestingly, it follows that graphs of large minimum degree contain clique-minors whose order is an exponential function of their girth. In particular, it follows by the main result of Kühn and Osthus in [21] that there is a constant c such that, if a graph has minimum degree d ≥ 3 and girth z, then it contains as a minor a clique of size k, whereIn this paper we provide conditions, alternative to the above one, that can force the existence of a clique-minor whose size is exponential.H-girth.…”