Colorability of mixed hypergraphs and their chromatic inversions

Abstract

We solve a long-standing open problem concerning a discrete mathematical model, which has various applications in computer science and several other fields, including frequency assignment and many other problems on resource allocation. A mixed hypergraph $\mathcal H$ is a triple $(X,\mathcal C ,\mathcal D )$, where $X$ is the set of vertices, and $\mathcal C$ and $\mathcal D$ are two set systems over $X$, the families of so-called C-edges and D-edges, respectively. A vertex coloring of a mixed hypergraph $\mathcal H$ is proper if every C-edge has two vertices with a common color and every D-edge has two vertices with different colors. A mixed hypergraph is colorable if it has at least one proper coloring; otherwise it is uncolorable. The chromatic inversion of a mixed hypergraph $\mathcal H =(X,\mathcal C ,\mathcal D )$ is defined as $\mathcal H ^c=(X,\mathcal D ,\mathcal C )$. Since 1995, it was an open problem wether there is a correlation between the colorability properties of a hypergraph and its chromatic inversion. In this paper we answer this question in the negative, proving that there exists no polynomial-time algorithm (provided that $P \ne NP$) to decide whether both $\mathcal H$ and $\mathcal H ^c$ are colorable, or both are uncolorable. This theorem holds already for the restricted class of 3-uniform mixed hypergraphs (i.e., where every edge has exactly three vertices). The proof is based on a new polynomial-time algorithm for coloring a special subclass of 3-uniform mixed hypergraphs. Implementation in C++ programming language has been tested. Further related decision problems are investigated, too.