On the Corrádi–Hajnal theorem and a question of Dirac

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Abstract

In 1963, Corrádi and Hajnal proved that for all k1 and n3k, 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)2k1 that contain k disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of (2k1)-connected graphs with no k disjoint cycles.

Enomoto and Wang refined the Corrádi–Hajnal Theorem, proving the following Ore-type version: For all k1 and n3k, every graph G on n vertices contains k disjoint cycles, provided that d(x)+d(y)4k1 for all distinct nonadjacent vertices x,y. We refine this further for k3 and n3k+1: If G is a graph on n vertices such that d(x)+d(y)4k3 for all distinct nonadjacent vertices x,y, then G has k vertex-disjoint cycles if and only if the independence number α(G)n2k 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.

Keywords

Disjoint cycles
Ore-degree
Graph packing
Equitable coloring
Minimum degree

Cited by (0)

The first two authors thank Institut Mittag-Leffler (Djursholm, Sweden) for the hospitality and creative environment.

1

Research of this author is supported in part by NSA grant H98230-12-1-0212.

2

Research of this author is supported in part by NSF grants DMS-1266016 and DMS-1600592 and by grant 12-01-00448 of the Russian Foundation for Basic Research.

3

Research of this author is supported in part by NSF grant DMS-1266016.