A Remark on Hamiltonian Cycles

Let G be an undirected and simple graph on n vertices. Let ω, α and χ denote the number of components, the independence number and the connectivity number of G. G is called a 1‐tough graph if ω( G – S ) ⩽ | S | for any subset S of V ( G ) such that ω( G − S ) > 1. Let σ 2 = min { d ( v ) + d ( w )| v and w are nonadjacent}. Note that the difference α ‐ χ in 1‐tough graph may be made arbitrary large. In this paper we prove that any 1‐tough graph with σ 2 > n + χ ‐ α is hamiltonian.