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arXiv · 2608.09895

Superlinear Lower Bounds for Monochromatic Path Partitions

Heng Li·Lanchao Wang
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Abstract

In 1989, Gy\'arf\'as conjectured that the vertex set of every $r$-edge-coloured complete graph can be partitioned into at most $r$ vertex-disjoint monochromatic paths. Erd\H{o}s, Gy\'arf\'as, and Pyber subsequently proposed the analogous conjecture for monochromatic cycles. Pokrovskiy proved Gy\'arf\'as's conjecture for $r=3$, while disproving the conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber for every $r\ge3$ by constructing colourings that require at least $r+1$ monochromatic cycles. In this paper, we disprove Gy\'arf\'as's conjecture in a quantitatively strong superlinear form: for every sufficiently large $r$, there exists an $r$-edge-coloured complete graph that requires at least $(1-o(1))r\log\log r$ vertex-disjoint monochromatic paths. Consequently, the monochromatic cycle-partition number is also superlinear in $r$. Our construction also disproves two conjectures of Pokrovskiy: one on monochromatic cycle coverings and the other on path coverings in the balanced bipartite setting.

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Heng Li, Lanchao Wang. 2026-08-10. Superlinear Lower Bounds for Monochromatic Path Partitions. https://arxiv.org/abs/2608.09895

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