Pancyclicity of 4-Connected, Claw-Free, P10-Free Graphs

A graph G is said to be pancyclic if G contains cycles of all lengths from 3 to *. We show that if G is 4-connected, claw-free, and P10-free, then G is either pancyclic or it is the line graph of the Petersen graph. This implies that every 4-connected, claw-free, P9-free graph is pancyclic, which is best possible and extends a result of Gould et al. Pancyclicity in 3-connected graphs: Pairs of forbidden subgraphs, [J Graph Theory 47 (2004), 183–202]. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; Contract grant number: 0742434; Contract grant sponsor: UCD GK-12 Transforming Experiences Project.This research was completed while the third author was an Assistant Research Professor at the University of Colorado, Denver.)