Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture for $P_{10}$-free Graphs

TL;DR

This paper proves that all $P_{10}$-free graphs with minimum degree at least three contain a cycle of length 4 or 8, supporting the Erdős-Gyárfás Conjecture within this class of graphs.

Contribution

It establishes the conjecture for $P_{10}$-free graphs by showing they contain cycles of length 4 or 8 when minimum degree is at least three.

Findings

Every $P_{10}$-free graph with minimum degree ≥ 3 contains a cycle of length 4 or 8.

Supports Erdős-Gyárfás Conjecture for a specific class of graphs.

Provides structural insights into $P_{10}$-free graphs.

Abstract

Let $P_{10}$ be a path on $10$ vertices. A graph is said to be $P_{10}$-free if it does not contain $P_{10}$ as an induced subgraph. The well-known Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of $2$. In this paper, we show that every $P_{10}$-free graph with minimum degree at least three contains a cycle of length $4$ or $8$. This implies that the conjecture is true for $P_{10}$-free graphs.