Erdős–Gyárfás conjecture for P8 -free graphs

Abstract

A graph is P8-free if it contains no induced subgraph isomorphic to the path P8 on eight vertices. In 1995, Erdős and Gyárfás conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two. In this paper, we confirm the conjecture for P8-free graphs by showing that there exists a cycle of length four or eight in every P8-free graph with minimum degree at least three.