Pairs of Fan-type heavy subgraphs for pancyclicity of 2-connected graphs

A graph G on n vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to n. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected graph to be Hamiltonian. Motivated by Fan’s result, we say that an induced subgraph H of G is f1-heavy if for every pair of vertices u, v ∈ V (H), dH(u, v) = 2 implies max{d(u), d(v)} ≥ (n + 1)/2. For a given graph R, G is called R-f1-heavy if every induced subgraph of G isomorphic to R is f1-heavy. In this paper we show that for a connected graph S with S = P3 and a 2-connected claw-f1-heavy graph G which is not a cycle, G being S-f1-heavy implies G is pancyclic if S = P4, Z1 or Z2, where claw is K1,3 and Zi is the path a1a2a3 . . . ai+2ai+3 plus the edge a1a3. Our result partially improves a previous theorem due to Bedrossian on pancyclicity of 2-connected graphs.