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 σ 2 ( G ) ≥ n is hamiltonian, where σ 2 ( G ) = min { d G ( x ) + d G ( y ) : x , y ∈ V ( G ) , x ≠ y , x y ∉ 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 2 n ≥ 4 with partite sets X and Y satisfying σ 1 , 1 ( H ) ≥ n , where σ 1 , 1 ( H ) = min { d H ( x ) + d H ( y ) : x ∈ X , y ∈ Y , x y ∉ 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.