A Note on Minimum Degree, Bipartite Holes, and Hamiltonian Properties

We adopt the recently introduced concept of the bipartite-hole-number due to McDiarmid and Yolov, and extend their result on Hamiltonicity to other Hamiltonian properties of graphs with a large minimum degree in terms of this concept. An (s, t)-bipartite-hole in a graph G consists of two disjoint sets of vertices S and T with |S| = s and |T| = t such that E(S, T ) = circle divide. The bipartite-hole-number (alpha) over tilde (G) is the maximum integer r such that G contains an (s, t)-bipartite-hole for every pair of nonnegative integers s and t with s + t = r. Our main results are that a graph G is traceable if delta(G) >= (alpha) over tilde (G) - 1, and Hamilton-connected if delta(G) >= (alpha) over tilde (G) + 1, both improving the analogues of Dirac's Theorem for traceable and Hamilton-connected graphs.