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

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.