Uncolorable mixed hypergraphs

Abstract

A mixed hypergraph H=(X, A,E) consists of the vertex set X and two families of subsets: the family E of edges and the family A of co-edges. In a coloring every edge E∈ E has at least two vertices of different colors, while every co-edge A∈ A has at least two vertices of the same color. The largest (smallest) number of colors for which there exists a coloring of a mixed hypergraph H using all the colors is called the upper (lower) chromatic number and is denoted χ ̄ ( H) (χ( H)) . A mixed hypergraph is called uncolorable if it admits no coloring. We show that there exist uncolorable mixed hypergraphs H=(X, A,E) with arbitrary difference between the upper chromatic number χ ̄ ( H A ) of H A =(X, A) and the lower chromatic number χ( H E ) of H E =(X, E). Moreover, for any k= χ ̄ ( H A )−χ( H E )⩾0 , the minimum number v( k) of vertices of an inclusionwise minimal uncolorable mixed hypergraph is exactly k+4. We introduce a measure of uncolorability (the vertex uncolorability number) and propose a greedy algorithm that finds an estimate on it. We also show that the colorability problem can be expressed in terms of integer programming. Concerning particular cases, we describe those complete ( l, m)-uniform mixed hypergraphs which are uncolorable, and observe that for any fixed ( l, m) almost all complete ( l, m)-uniform mixed hypergraphs are uncolorable, whereas generally almost all complete mixed hypergraphs are colorable.