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arXiv · 2603.04704

Covering complete $r$-partite hypergraphs with few monochromatic components

Luke Hawranick·Ruth Luo
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Abstract

An edge-coloring of a hypergraph is {\em spanning} if every vertex sees every color used in the coloring. In this paper, we prove that for $k \geq 2r \geq 6$, in any spanning $k$-coloring of the edges of a complete $r$-partite $r$-uniform hypergraph $H$, the vertices of $H$ can be covered by a set of at most $k-r+1$ monochromatic connected components. This proves a conjecture of Gyárfás and Király which is related to a special case of Ryser's conjecture. We also prove that for $k \in \{2,3\}$, every spanning $k$-edge-coloring of a complete bipartite graph admits a covering of its vertices using at most $k$ monochromatic components.

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Luke Hawranick, Ruth Luo. 2026-03-05. Covering complete $r$-partite hypergraphs with few monochromatic components. https://arxiv.org/abs/2603.04704

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