Disjoint cycles and 2-factors with Fan-type condition in a graph

Abstract

Let k be a positive integer. For a graph G, let σ12(G) be the minimum value of max{d(x),d(y)} for any pair of nonadjacent vertices x and y. Corrádi and Hajnal proved that every graph G of order n≥3k with δ(G)≥2k contains k disjoint cycles. In this paper, we generalize this theorem by proving that, for a graph G of order at least 3k+2, if σ12(G)≥2k, then G contains k disjoint cycles. The bound of order is sharp. Based on this result, we show that if G is a 2-connected graph of order at least 3k+5 with σ12(G)≥max{n/2,2k+2}, then G contains a 2-factor with exactly k disjoint cycles, which improves the theorem of Brandt et al. (1997) [2].