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arXiv · 2608.27331

Sharp Same-Color Cycle Covers in Two-Colored Complete Graphs

Xiao-Chuan Liu·Jonatas Teodomiro·Xu Yang
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Abstract

We extend the conjecture of Erdős and Gyárfás on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, that together cover all vertices. The order of the bound is best possible, and the ceiling is necessary for infinitely many $n$.

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Xiao-Chuan Liu, Jonatas Teodomiro, Xu Yang. 2026-08-27. Sharp Same-Color Cycle Covers in Two-Colored Complete Graphs. https://arxiv.org/abs/2608.27331

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