Hamilton-connected, vertex-pancyclic and bipartite holes

Abstract

In 2016, McDiarmid and Yolov gave a tight threshold for the existence of Hamilton cycle in graphs with large minimum degree and without large “bipartite hole”(two disjoint sets of vertices with no edge between them) which extends Dirac's classical Theorem. In detail, an (s,t)-bipartite-hole in a graph G consists of two disjoint vertex sets S and T with |S|=s and |T|=t such that E(G[S,T])=∅. Let α˜(G) be the maximum integer such that G contains an (s,t)-bipartite-hole for every pair of nonnegative integers s and t with s+t=r. Motivated by Bondy's metaconjecture, in this paper, we study the existence of vertex-pancyclicity (every vertex is in a cycle of length i for each i∈[3,n] and Hamilton-connectivity(any two vertices can be connected through a Hamilton path). Our central theorem is that for any given μ>0 and sufficiently large n, if G is an n-vertex graph with α˜(G)=μn and δ(G)≥103μn, then G is Hamilton-connected and vertex-pancyclic.