Pancyclicity of 4-Connected $$\{K_{1,3},Z_8\}$${K1,3,Z8}-Free Graphs

AbstractA graph G is said to be pancyclic if G contains cycles of lengths from 3 to |V(G)|. For a positive integer i, we use $$Z_i$$Zi to denote the graph obtained by identifying an endpoint of the path $$P_{i+1}$$Pi+1 with a vertex of a triangle. In this paper, we show that every 4-connected claw-free $$Z_8$$Z8-free graph is either pancyclic or is the line graph of the Petersen graph. This implies that every 4-connected claw-free $$Z_6$$Z6-free graph is pancyclic, and every 5-connected claw-free $$Z_8$$Z8-free graph is pancyclic.