An Ore-type condition for existence of two disjoint cycles

Abstract

Let $n_1$ and $n_2$ be two integers with $n_1,n_2\ge 3$ and $G$ a graph of order $n=n_1+n_2$. As a generalization of Ore's degree condition for the existence of Hamilton cycle in $G$, El-Zahar proved that if $\delta (G)\ge \lceil \frac{n_1}{2}\rceil +\lceil \frac{n_2}{2}\rceil$ then $G$ contains two disjoint cycles of length $n_1$ and $n_2$. Recently, Yan et. al considered the problem by extending the degree condition to degree sum condition and proved that if $d(u)+d(v)\ge n+4$ for any pair of non-adjacent vertices $u$ and $v$ of $G$, then $G$ contains two disjoint cycles of length $n_1$ and $n_2$. They further asked whether the degree sum condition can be improved to $d(u)+d(v)\ge n+2$. In this paper, we give a positive answer to this question. Our result also generalizes El-Zahar's result when $n_1$ and $n_2$ are both odd.