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

Luke Hawranick, Ruth Luo

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.

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

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 , in any spanning -coloring of the edges of a complete -partite -uniform hypergraph , the vertices of can be covered by a set of at most 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 , every spanning -edge-coloring of a complete bipartite graph admits a covering of its vertices using at most monochromatic components.
Paper Structure (6 sections, 5 theorems, 8 equations, 2 figures)

This paper contains 6 sections, 5 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spanning colorings and new results
  3. Proof of Theorem \ref{['main']}
  4. The case $r=3$
  5. The case $r \ge 4$
  6. Proof of Theorem \ref{['cov2k']}

Key Result

Theorem 1.4

If $H$ is a $k$-edge-colored complete $r$-uniform hypergraph with $r \geq 3$, then at most $\left\lceil k/r \right\rceil$ monochromatic components are needed to cover $V(H)$, and this bound is sharp.

Figures (2)

  • Figure 1: Components for colors $c_1$ and $c_2$ used in the proof of case $r=3$.
  • Figure 2: Vectors used in the proof of case $r \geq 4$.

Theorems & Definitions (28)

  • Conjecture 1.1: Henderson Henderson, attributed to Ryser
  • Conjecture 1.2: Gyárfás Gyarfas
  • Conjecture 1.3: GyarfasLehel
  • Theorem 1.4: Király Kiraly
  • Theorem 1.5: Gyárfás, Király GK
  • Conjecture 1.6: Gyárfás, Király GK
  • Theorem 1.7
  • Theorem 1.8
  • Claim 2.1
  • proof
  • ...and 18 more