A sufficient condition for a graph to contain two disjoint Hamiltonian cycles (Q915741)

There is proved the following main result which generalizes Ore's theorem: Let G be a graph with n vertices and let \(d_ G(x)\) denote the degree of the vertex x in G; if \(n\geq 5\) and \(d_ G(x)+d_ G(y)\geq n+2\) holds for all nonadjacent vertices \(x\neq y\) of G, then G contains two disjoint Hamiltonian cycles. The proof is based on Ore's theorem and three lemmata.