Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Abstract

Given a function f : N → R, call an n-vertex graph f -connected if separating off k vertices requires the deletion of at least f (k) vertices whenever k ≤ (n − f (k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f -connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f (k) ≥ 2k + 1, and contains a Hamilton cycle if f (k) ≥ 2(k + 1) 2 . We conjecture that linear growth of f suffices to imply hamiltonicity.

Hamilton cycles in highly connected and expanding graphs

“…We would like to thank Deryk Osthus for suggesting to use our criterion to address the conjecture of [5].…”

Matchings in regular graphs from eigenvalues

“…For, it is not difficult to check that if ν(G) n−r+1 2 in the proof of Lemma 5, then G has r vertex disjoint induced subgraphs H i ∈ H(k) (see also [8,Theorem 11]), and so λ r (G) ρ(k) as in (2). Thus (7) implies that bounds on lower eigenvalues guarantee the existence of smaller matchings.…”

“…If G is a k-regular Ramanujan graph of even order n with k 6, then λ 2 (G) 2 Every k-regular graph G with k 3 and with diameter at most 3 must have ν(G) = n/2 . For if ν(G) < n/2 , then G must have diameter greater than 3 because (as noted in [3, Corollary 3.4]), t i < n i for i ∈ {1, 2, 3} in the proof of Lemma 5.…”

The size Ramsey number of a directed path

“…The following lemma shows that every k-pseudorandom graph contains a long path. The proof follows ideas from [7,5]. Proof.…”