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) = θ. The bipartite-hole-number ∼α(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 δ(G) ≥ ∼α(G)-1, and Hamilton-connected if δ(G) ≥ ∼;α(G) + 1, both improving the analogues of Dirac's Theorem for traceable and Hamilton-connected graphs.