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

TL;DR

This paper proves that in certain complete hypergraphs with spanning edge-colorings, the vertices can be covered by a limited number of monochromatic connected components, confirming a conjecture related to Ryser's conjecture.

Contribution

It establishes a bound on the number of monochromatic components needed to cover vertices in spanning colorings of complete hypergraphs, confirming a conjecture of Gyárfás and Király.

Findings

For $k \\geq 2r \\geq 6$, vertices can be covered by at most $k-r+1$ monochromatic components.

For $k \\in \\{2,3\\}$, the vertices of a complete bipartite graph can be covered by at most $k$ monochromatic components.

Proves a special case of Ryser's conjecture related to hypergraph colorings.

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 Gy\'arf\'as and Kir\'aly 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.