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

Let k be an integer greater than one, and let G be a simple graph with at least 4k+1 vertices. In this article, we prove that if σ2(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 σ2(G) is the minimum degree sum for a pair of non-adjacent vertices.