2016/01/15 by Kierstead, H. A., Kostochka, A. V., Yeager, E. C.
#05C15 #05C35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1601.03791
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.