On a 2-factor with a specified edge in a graph satisfying the Ore condition

Abstract

Let k be an integer greater than one, and let G be a simple graph with at least 4 k+1 vertices. In this article, we prove that if σ 2( G)⩾| V( G)|, then for an edge e of G, there exists a 2-factor with k cycles that contains e, or | V( G)| is even and G has a vertex cover of size | V( G)|/2 containing the endpoints of e. Here σ 2( G) is the minimum degree sum for a pair of non-adjacent vertices.