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arXiv · 2308.05675

Erdős-Gyárfás Conjecture for $P_{10}$-free Graphs

Zhiquan Hu·Changlong Shen
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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 Erdős-Gyárfás 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.

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Zhiquan Hu, Changlong Shen. 2023-08-12. Erdős-Gyárfás Conjecture for $P_{10}$-free Graphs. https://arxiv.org/abs/2308.05675

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