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’.