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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, Fanpresented a degree condition involving every pair of vertices at distance twofor a 2-connected graph to be Hamiltonian. Motivated by Fan's result, we saythat an induced subgraph $H$ of $G$ is $f_1$-heavy if for every pair ofvertices $u,v\in V(H)$, $d_{H}(u,v)=2$ implies $\max\{d(u),d(v)\}\geq (n+1)/2$.For a given graph $R$, $G$ is called $R$-$f_1$-heavy if every induced subgraphof $G$ isomorphic to $R$ is $f_1$-heavy. In this paper we show that for aconnected graph $S$ with $S\neq P_3$ and a 2-connected claw-$f_1$-heavy graph$G$ which is not a cycle, $G$ being $S$-$f_1$-heavy implies $G$ is pancyclic if$S=P_4,Z_1$ or $Z_2$, where claw is $K_{1,3}$ and $Z_i$ is the path$a_1a_2a_3... a_{i+2}a_{i+3}$ plus the edge $a_1a_3$. Our result partiallyimproves a previous theorem due to Bedrossian on pancyclicity of 2-connectedgraphs.

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