On proper colorings of hypergraphs

TL;DR

This paper establishes new upper bounds on the number of colors needed for proper vertex coloring of hypergraphs based on their maximum degree and edge size, improving understanding of coloring constraints.

Contribution

It proves that hypergraphs with certain degree and edge size conditions can be properly colored with fewer colors than previously known.

Findings

Hypergraphs with degree $ riangle$ and edge size $oldsymbol{ ext{ extdelta}}$ are colorable with $oldsymbol{k+1}$ colors.

If $ extdelta extgreater= 3$ and $k extgreater= 3$, then hypergraphs are colorable with $oldsymbol{k}$ colors.

Derived upper bounds on colors needed for dynamic colorings of hypergraphs.

Abstract

Let $\mathcal{H}$ be a hypergraph of maximal vertex degree $\Delta$, such that each its hyperedge contains at least $\delta$ vertices. Let $k=\lceil\frac{2\Delta}{\delta}\rceil$. We prove that (i) The hypergraph $\mathcal{H}$ admits proper vertex coloring in $k+1$ colors. (ii) The hypergraph $\mathcal{H}$ admits proper vertex coloring in $k$ colors, if $\delta\ge 3$ and $k\ge 3$. As a consequence of these results we derive upper bounds on the number of colors in dynamic colorings.