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

Abstract 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 ) =∅. The bipartite-hole-number α˜(G) \tilde \alpha \left( G \right) 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 \delta \left( G \right) \ge \tilde \alpha \left( G \right) - 1 , and Hamilton-connected if δ(G)≥α˜(G)+1 \delta \left( G \right) \ge \tilde \alpha \left( G \right) + 1 , both improving the analogues of Dirac’s Theorem for traceable and Hamilton-connected graphs.