Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

TL;DR

This paper develops spectral bounds for various hypergraph coloring problems, extending known inequalities and identifying conditions for tightness, thereby advancing understanding of hypergraph spectral properties in coloring theory.

Contribution

It generalizes spectral bounds to $d$-proper and $d$-improper hypergraph colorings and provides necessary conditions for these bounds to be tight.

Findings

Derived spectral bounds for $d$-proper hypergraph colorings.

Extended bounds to $d$-improper colorings and edge colorings.

Identified conditions for the bounds to be tight.

Abstract

It is known that, for an oriented hypergraph with (vertex) coloring number $\chi$ and smallest and largest normalized Laplacian eigenvalues $\lambda_1$ and $\lambda_N$, respectively, the inequality $\chi\geq (\lambda_N-\lambda_1)/\min\{\lambda_N-1,1-\lambda_1\}$ holds. We provide necessary conditions for oriented hypergraphs for which this bound is tight. Focusing on $c$-uniform unoriented hypergraphs, we then generalize the bound to the setting of \emph{$d$-proper colorings}: colorings in which no edge contains more than $d$ vertices of the same color. We also adapt our proof techniques to derive analogous spectral bounds for \emph{$d$-improper colorings} of graphs and for edge colorings of hypergraphs. Moreover, for all coloring notions considered, we provide necessary conditions under which the bound is an equality.