Hypergraph Colouring and Degeneracy

TL;DR

This paper investigates the chromatic properties of hypergraphs with degeneracy constraints, establishing tight bounds on coloring and constructing specific examples that meet these bounds.

Contribution

It proves the tightness of the greedy coloring bound for $d$-degenerate hypergraphs and constructs triangle-free hypergraphs with maximal chromatic number.

Findings

The greedy algorithm uses at most $d+1$ colors for $d$-degenerate hypergraphs.

Constructed hypergraphs are $r$-uniform, $d$-degenerate, and have chromatic number $d+1$.

Hypergraphs can be triangle-free yet require $d+1$ colors.

Abstract

A hypergraph is "$d$-degenerate" if every subhypergraph has a vertex of degree at most $d$. A greedy algorithm colours every such hypergraph with at most $d+1$ colours. We show that this bound is tight, by constructing an $r$-uniform $d$-degenerate hypergraph with chromatic number $d+1$ for all $r\geq2$ and $d\geq1$. Moreover, the hypergraph is triangle-free, where a "triangle" in an $r$-uniform hypergraph consists of three edges whose union is a set of $r+1$ vertices.