An Ore-type condition for the existence of two disjoint cycles

Abstract

Let n1 and n2 be two integers with n1,n2≥3 and G a graph of order n=n1+n2. As a variation of Ore's degree condition for the existence of Hamilton cycle in G, El-Zahar proved that if δ(G)≥⌈n12⌉+⌈n22⌉, then G contains two disjoint cycles of length n1 and n2. Recently, Yan et al. considered the problem by extending the degree condition to degree sum condition and proved that if d(u)+d(v)≥n+4 for any pair of non-adjacent vertices u and v of G, then G contains two disjoint cycles of length n1 and n2. They further asked whether the degree sum condition can be improved to d(u)+d(v)≥n+2. In this paper, we give a positive answer to this question. Our result also extends El-Zahar's result when n1 and n2 are both odd.