Small cycles in Hamiltonian graphs

Abstract

We prove that if a graph G on n ⩾ 32 vertices is hamiltonian and has two nonadjacent vertices u and v with d(u) + d(v) ⩾ n + z where z = 0 if n is odd and z = 1 if n is even, then G contains all cycles of length m where $\text{3 ⩽ m ⩽}\text{1}\text{5}\text{(n + 13)}$.