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arXiv · 2211.03291

Rainbow cycles in properly edge-colored graphs

Jaehoon Kim·Joonkyung Lee·Hong Liu·Tuan Tran
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Abstract

We prove that every properly edge-colored $n$-vertex graph with average degree at least $100(\log n)^2$ contains a rainbow cycle, improving upon $(\log n)^{2+o(1)}$ bound due to Tomon. We also prove that every properly colored $n$-vertex graph with at least $10^5 k^2 n^{1+1/k}$ edges contains a rainbow $2k$-cycle, which improves the previous bound $2^{ck^2}n^{1+1/k}$ obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erd\H{o}s--Simonovits supersaturation theorem for even cycles, which may be of independent interest.

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Jaehoon Kim, Joonkyung Lee, Hong Liu, Tuan Tran. 2022-11-07. Rainbow cycles in properly edge-colored graphs. https://arxiv.org/abs/2211.03291

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