Mixed interval hypergraphs

Abstract

We investigate the coloring properties of mixed interval hypergraphs having two families of subsets: the edges and the co-edges. In every edge at least two vertices have different colors. The notion of a co-edge was introduced recently in Voloshin (1993, 1995): in every such a subset at least two vertices have the same color. The upper (lower) chromatic number is defined as a maximum (minimum) number of colors for which there exists a coloring of a mixed hypergraph using all the colors. We find that for colorable mixed interval hypergraph H the lower chromatic number χ( H) ⩽ 2, the upper chromatic number χ(H) = |X|−s(H) , where s( H) is introduced as the so-called sieve number. A characterization of uncolorability of a mixed interval hypergraph is found, namely: such a hypergraph is uncolorable if and only if it contains an obviously uncolorable edge. The co-stability number α.√( H) is the maximum cardinality of a subset of vertices which contains no co-edge. A mixed hypergraph H is called co-perfect if χ(H′) = α√(H′) for every subhypergraph H′. Such minimal non-co-perfect hypergraphs as monostars and cycloids C r 2 r−1 have been found in Voloshin (1995). A new class of non-co-perfect mixed hypergraphs called covered co-bi-stars is found in this paper. It is shown that mixed interval hypergraphs are coperfect if and only if they do not contain co-monostars and covered co-bi-stars as subhypergraphs. Linear time algorithms for computing lower and upper chromatic numbers and respective colorings for this class of hypergraphs are suggested.