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arXiv · 2410.22842

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

Anand Shripad Hegde·R. B. Sandeep·P. Shashank
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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.

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Anand Shripad Hegde, R. B. Sandeep, P. Shashank. 2024-10-30. Erd\H{o}s-Gy\'arf\'as conjecture on graphs without long induced paths. https://arxiv.org/abs/2410.22842

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