Spanning bipartite graphs with high degree sum in graphs

The classical Ore's Theorem states that every graph G of order n >= 3 with sigma(2)(G) >= n is hamiltonian, where sigma(2)(G) = min{d(G)(x) + d(G)(y): x, y is an element of V(G), x not equal y, xy is not an element of 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 sigma(1,1)(H) >= n, where sigma(1.1)(H) = min{d(H)(x)+d(H)(y): x is an element of X, y is an element of Y, xy is not an element of 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. (C) 2019 Elsevier B.V. All rights reserved.