Clique-heavy subgraphs and pancyclicity of 2-connected graphs

For a given graph H we say that G is H-c1-heavy if for every induced subgraph K of G isomorphic to H and every maximal clique C in K there is a super-heavy vertex in every non-trivial component of K−C; and that G is H-o1-heavy if in every induced subgraph of G isomorphic to H there are two non-adjacent vertices with sum of degrees at least |G|+1. Let Z1 denote a graph consisting of a triangle with a pendant edge. In this paper we prove that every 2-connected K1,3-o1-heavy and Z1-c1-heavy graph is pancyclic. As a consequence we obtain a complete characterization of graphs H such that every 2-connected graph claw-o1-heavy and H-c1-heavy graph is pancyclic. This result extends previous work by Bedrossian [1].