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.