Defective Colouring of Hypergraphs

TL;DR

This paper generalizes a classical hypergraph coloring result by showing that vertices can be colored with a specific number of colors so that each vertex belongs to at most a certain number of monochromatic edges, extending the case where no monochromatic edges are allowed.

Contribution

It introduces a new bound for defective coloring of hypergraphs, extending the Erdős-Lovász result to cases allowing some monochromatic edges.

Findings

Optimal coloring bounds up to a constant factor

Extension of classical hypergraph coloring results

Applicable to hypergraphs with bounded maximum degree

Abstract

We prove that the vertices of every $(r + 1)$-uniform hypergraph with maximum degree $\Delta$ may be coloured with $c(\frac{\Delta}{d + 1})^{1/r}$ colours such that each vertex is in at most $d$ monochromatic edges. This result, which is best possible up to the value of the constant $c$, generalises the classical result of Erd\H{o}s and Lov\'asz who proved the $d = 0$ case.