A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs

A graph G on n vertices is said to be pancyclic if it contains cycles of all lengths k for k ∈ {3, . . ., n}. A vertex v ∈ V (G) is called super-heavy if the number of its neighbours in G is at least (n+1)/2. For a given graph H we say that G is H-f₁-heavy if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), d_K(u, v) = 2 implies that at least one of them is super-heavy. For a family of graphs ℋ we say that G is ℋ-f₁-heavy, if G is H-f₁-heavy for every graph H ∈ℋ. Let D denote the deer, a graph consisting of a triangle with two disjoint paths P3 adjoined to two of its vertices. In this paper we prove that every 2-connected {K_1,3, P₇, D}-f₁-heavy graph on n ≥ 14 vertices is pancyclic. This result extends the previous work by Faudree, Ryjáček and Schiermeyer.