On n-hamiltonian graphs

Abstract

A graph G of order p ⩾ 3 is called n-hamiltonian, 0 ⩽ n ⩽ p − 3, if the removal of any m vertices, 0 ⩽ m ⩽ n, results in a hamiltonian graph. A graph G of order p ⩾ 3 is defined to be n-hamiltonian, − p ⩽ n ⩽ 1, if there exists − n or fewer pairwise disjoint paths in G which collectively span G. Various conditions in terms of n and the degrees of the vertices of a graph are shown to be sufficient for the graph to be n-hamiltonian for all possible values of n. It is also shown that if G is a graph of order p ⩾ 3 and K(G) ⩾ β(G) + n (−p ⩽ n ⩽ p − 3) , then G is n-hamiltonian.