A Fan-type heavy pair od subgraphs for pancyclicity of 2-connected graphs

Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it is H-f1-heavy, if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies min⁡{dG(u),dG(v)}≥n+12$\min \{ d_G (u),d_G (v)\} \ge {{n + 1} \over 2}$. In this paper we prove that every 2-connected {K1,3, P5}-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying pancyclicity of 2-connected graphs