Cycles containing prescribed vertices

Abstract

Let G be a graph and S a prescribed subset of the vertex set of G. Conditions are given which guarantee that there there is a cycle or path in G which contains the vertices of S. These results extend Ore's theorem on the hamiltonian properties of a graph and Gould's theorems on the hamiltonian properties of a claw-free graph. The Nash-Williams' theorem on the hamiltonicity of a graph is also generalized.