On regular 2-connected 2-path Hamiltonian graphs

Abstract

A graph is l-path Hamiltonian if every path of length not exceeding l is contained in a Hamiltonian cycle. It is well known that a 2-connected, k-regular graph G on at most 3k−1 vertices is edge-Hamiltonian if for every edge uv of G, {u,v} is not a cut-set. Thus G is 1-path Hamiltonian if G−{u,v} is connected for every edge uv of G. Let P=uvz be a 2-path of a 2-connected, k-regular graph G on at most 2k vertices. In this paper, we show that there is a Hamiltonian cycle containing the 2-path P if G−V(P) is connected. Therefore, the work implies a condition for a 2-connected, k-regular graph to be 2-path Hamiltonian. An example shows that the 2k is almost sharp, i.e., the number is at most 2k+1.