Two edge-disjoint Hamiltonian cycles in graphs (Q1334943)

It is shown that if a graph is 2-connected of order \(n\geq 40\) with minimum degree of the graph \(\geq 7\) and \(d(u)+ d(v)\geq n-1\) for every pair of nonadjacent vertices \(u\) and \(v\), then the graph has two edge- disjoint Hamiltonian cycles unless it is one of two special graphs, and this conclusion is best possible.