Erd\H{o}s-Gy\'arf\'as conjecture on graphs without long induced paths

TL;DR

This paper advances the Erd ext{"o}s-Gy ext{"a}rf ext{"a}s conjecture by proving it holds for graphs without induced paths of length 13, extending previous results for shorter paths using computational methods.

Contribution

The paper extends the validity of the Erd ext{"o}s-Gy ext{"a}rf ext{"a}s conjecture to $P_{13}$-free graphs through computational proof, surpassing prior bounds.

Findings

Proves the conjecture for $P_{13}$-free graphs.

Uses computer-assisted search to verify the conjecture.

Builds on previous results for $P_8$-free and $P_{10}$-free graphs.

Abstract

Erd\H{o}s and Gy\'arf\'as conjectured in 1994 that every graph with minimum degree at least 3 has a cycle of length a power of 2. In 2022, Gao and Shan (Graphs and Combinatorics) proved that the conjecture is true for $P_8$-free graphs, i.e., graphs without any induced copies of a path on 8 vertices. In 2024, Hu and Shen (Discrete Mathematics) improved this result by proving that the conjecture is true for $P_{10}$ -free graphs. With the aid of a computer search, we improve this further by proving that the conjecture is true for $P_{13}$ -free graphs.