Spectrum of mixed bi-uniform hypergraphs

TL;DR

This paper characterizes the feasible color sets of mixed hypergraphs with edges of fixed sizes and constructs hypergraphs with prescribed numbers of proper colorings, extending previous results with a shorter proof.

Contribution

It provides a complete characterization of feasible sets for mixed hypergraphs with edges of two fixed sizes and constructs hypergraphs with exactly specified numbers of proper colorings.

Findings

Complete characterization of feasible sets for mixed hypergraphs with all C-edges of size ℓ and D-edges of size m.

Construction of hypergraphs with exactly r(s) proper colorings for each s in [ℓ, n].

Generalization of previous results with a shorter proof.

Abstract

A mixed hypergraph is a triple $H=(V,\mathcal{C},\mathcal{D})$, where $V$ is a set of vertices, $\mathcal{C}$ and $\mathcal{D}$ are sets of hyperedges. A vertex-coloring of $H$ is proper if $C$-edges are not totally multicolored and $D$-edges are not monochromatic. The feasible set $S(H)$ of $H$ is the set of all integers, $s$, such that $H$ has a proper coloring with $s$ colors. Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a characterization of feasible sets for mixed hypergraphs with all $C$- and $D$-edges of the same size $r$, $r\geq 3$. In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all $C$-edges of size $\ell$ and all $D$-edges of size $m$, where $\ell, m \geq 2$. Moreover, we show that for every sequence $(r(s))_{s=\ell}^n$, $n \geq \ell$, of natural numbers there exists such a hypergraph with exactly $r(s)$ proper colorings using $s$ colors, $s = \ell,\ldots,n$, and no proper coloring with more than $n$ colors. Choosing $\ell = m=r$ this answers a question of Bujt\'as and Tuza, and generalizes their result with a shorter proof.