Uniform Mixed Hypergraphs: The Possible Numbers of Colors

Abstract

For a mixed hypergraph $${\mathcal{H}}=(X,{\mathcal{C}},{\mathcal{D}})$$, where $$\mathcal{C}$$ and $$\mathcal{D}$$ are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’ such that every $$C \in {\mathcal{C}}$$ meets some class in more than one vertex, and every $$D \in {\mathcal{D}}$$ has a nonempty intersection with at least two classes. The feasible set of $${\mathcal{H}}$$, denoted $$\Phi({\mathcal{H}})$$, is the set of integers k such that $${\mathcal{H}}$$ admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309–318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either $$1 \notin S$$ or S is an ‘interval’ $$\{1, \ldots, k \}$$ for some integer k ≥ 1. In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all $$C \in {\mathcal{C}}$$ and all $$D \in {\mathcal{D}}$$, r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either $$S=\{1, \ldots, k \}$$ for some natural number k ≥ r − 1, or S is of the form $$S = S^\prime\cup S^{\prime\prime}$$ where S′′ is any (possibly empty) subset of $$\{r,\,r+1,\ldots\}$$ and S′ is either the empty set or {r − 1} or an ‘interval’ {k, k + 1, ..., r − 1} for some k, 2 ≤ k ≤ r − 2. We also prove that all these feasible sets $$S\not\ni 1$$ can be obtained under the restriction $${\mathcal{C}}={\mathcal{D}}$$, i.e. within the class of ‘bi-hypergraphs’.