Summary: Graph \(G\) on \(n\) vertices is said to be pancyclic if it contains cycles of all lengths \(k\) for \(k\in\{3,\dots,n\}\). A vertex \(v\in V(G)\) is called super-heavy if the number of its neighbours in \(G\) is at least \((n+1)/2\). The complete bipartite graph \(K_{1, 3}\) is called a claw.
For a given graph \(H\) we say that \(G\) is \(H\)-\(\mathrm c_1\)-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 \(H\)-\(\mathrm o_1\)-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 \(Z_1\) denote a graph consisting of a triangle with a pendant edge.
In this paper we prove that every 2-connected \(K_{1, 3}\)-\(\mathrm o_1\)-heavy and \(Z_1\)-\(\mathrm c_1\)-heavy graph is pancyclic. As a consequence we obtain a complete characterization of graphs \(H\) such that every 2-connected graph claw-\(\mathrm o_1\)-heavy and \(H\)-\(\mathrm c_1\)-heavy graph is pancyclic. This result extends previous work by P. Bedrossian [Forbidden subgraph and minimum degree conditions for Hamiltonicity. Memphis, TN: Memphis State University (PhD Thesis) (1991)].