Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

It is known that, for an oriented hypergraph with (vertex) coloring number $χ$ and smallest and largest normalized Laplacian eigenvalues $λ_1$ and $λ_N$, respectively, the inequality $χ\geq (λ_N-λ_1)/\min{λ_N-1,1-λ_1}$ 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.

💡 Research Summary

This paper investigates spectral lower bounds for several hypergraph coloring problems by exploiting the eigenvalues of the normalized Laplacian. It begins with an alternative proof of the known bound for oriented hypergraphs, namely
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