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

A graph G is pancyclic if it contains cycles of each length ℓ , 3 ≤ ℓ ≤ | V ( G ) | . 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.