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arXiv · 2607.03243

On Ramsey-type problems for paths and cycles with few colour changes

Peter Allen·Julia Böttcher·Dennis Clemens·Fabian Hamann·Jozef Skokan·Anusch Taraz
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Abstract

In 1967, Gerencser and Gy\'arf\'as determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every $2$-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in $3$-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For $q,k \in \mathbb{N}$ and a graph $G$, let $R_q^k(G)$ denote the smallest $N \in \mathbb{N}$ such that every $q$-edge-coloured complete graph on $N$ vertices contains a copy of $G$ with at most $k$ vertices that are incident to edges in $G$ of different colours. For paths, we show that $R_3^1(P_n) = \frac{3n}{2} + O(1)$, and for even cycles, we show that $R_3^2(C_n) = \frac{3n}{2} + o(n)$.

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Peter Allen, Julia Böttcher, Dennis Clemens, Fabian Hamann, Jozef Skokan, Anusch Taraz. 2026-07-03. On Ramsey-type problems for paths and cycles with few colour changes. https://arxiv.org/abs/2607.03243

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