Spectrum of Mixed Bi-uniform Hypergraphs

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. Bujtás and Tuza (Graphs Combin 24:1–12, 2008) gave a characterization of feasible sets for mixed hypergraphs with all C- and D-edges of the same size $$r, r\ge 3$$ . 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 $$\ell , m \ge 2$$ . Moreover, we show that for every sequence $$(r(s))_{s=\ell }^n, n \ge \ell $$ , 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 Bujtás and Tuza, and generalizes their result with a shorter proof.