Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture for $P_8$-free graphs

TL;DR

This paper proves that in all $P_8$-free graphs with minimum degree at least three, there exists a cycle of length four or eight, confirming a conjecture by Erdős and Gyárfás for this class of graphs.

Contribution

The paper confirms the Erdős-Gyárfás conjecture for $P_8$-free graphs by demonstrating the existence of specific cycles in such graphs.

Findings

Existence of a 4-cycle or 8-cycle in $P_8$-free graphs with minimum degree ≥ 3

Verification of Erdős-Gyárfás conjecture for a new class of graphs

Structural properties of $P_8$-free graphs related to cycle lengths

Abstract

A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s 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 $P_8$-free graphs by showing that there exists a cycle of length four or eight in every $P_8$-free graph with minimum degree at least three.