Pancyclicity of 4-connected {claw, generalized bull}-free graphs

Abstract

A graph $G$ is pancyclic if it contains cycles of each length $\ell$, $3\le \ell \le \left|V\left(G\right)\right|$. The generalized bull $B(i,j)$ is obtained by associating one endpoint of each of the paths $P_{i+1}$ and $P_{j+1}$ with distinct vertices of a triangle. Gould, Łuczak and Pfender (2004) [4] showed that if $G$ is a 3-connected $\{K_{1,3},B(i,j)\}$-free graph with $i+j=4$ then $G$ is pancyclic. In this paper, we prove that every 4-connected, claw-free, $B(i,j)$-free graph with $i+j=6$ is pancyclic. As the line graph of the Petersen graph is $B(i,j)$-free for any $i+j=7$ and is not pancyclic, this result is best possible.