Spanning bipartite graphs with high degree sum in graphs

Abstract

The classical Ore’s Theorem states that every graph G of order n≥3 with σ2(G)≥n is hamiltonian, where σ2(G)=min{dG(x)+dG(y):x,y∈V(G),x≠y,xy∉E(G)}. Recently, Ferrara, Jacobson and Powell (Discrete Math. 312 (2012), 459–461) extended the Moon–Moser Theorem and characterized the non-hamiltonian balanced bipartite graphs H of order 2n≥4 with partite sets X and Y satisfying σ1,1(H)≥n, where σ1,1(H)=min{dH(x)+dH(y):x∈X,y∈Y,xy∉E(H)}. Though the latter result apparently deals with a narrower class of graphs, we prove in this paper that it implies Ore’s Theorem for graphs of even order.