On the Corrádi–Hajnal theorem and a question of Dirac

Abstract

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The bound δ(G)≥2k is sharp. Here we characterize those graphs with δ(G)≥2k−1 that contain k disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of (2k−1)-connected graphs with no k disjoint cycles.Enomoto and Wang refined the Corrádi–Hajnal Theorem, proving the following Ore-type version: For all k≥1 and n≥3k, every graph G on n vertices contains k disjoint cycles, provided that d(x)+d(y)≥4k−1 for all distinct nonadjacent vertices x,y. We refine this further for k≥3 and n≥3k+1: If G is a graph on n vertices such that d(x)+d(y)≥4k−3 for all distinct nonadjacent vertices x,y, then G has k vertex-disjoint cycles if and only if the independence number α(G)≤n−2k and G is not one of two small exceptions in the case k=3. We also show how the case k=2 follows from Lovász' characterization of multigraphs with no two disjoint cycles.