Hamiltonian Cycle Properties in k-Extendable Non-bipartite Graphs with High Connectivity

Abstract

Let G be a graph, $\nu$ the order of G and k a positive integer such that $k\le (\nu -2)/2$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its vertices, it contains a spanning path joining the two vertices. In this paper, we discuss k-extendable nonbipartite graphs with $\kappa (G)\ge 2k+r$ where $k\ge 1$ and $r\ge 0$. It is shown that if $\nu \le 6k+2r$, then G is Hamiltonian; and if $\nu > 6k+2r$, then G has a longest cycle C such that $|V(C)|\ge 6k+2r$; and if $\nu <6k+2r$, then G is Hamiltonian-connected; and if $\nu \ge 6k+2r$, then for each pair of vertices $z_1$ and $z_2$ of G, there is a path P of G joining $z_1$ and $z_2$ such that $|V(P)|\ge 6k+2r-2$. All the bounds are sharp and all results can be extended to 2k-factor-critical graphs.