Mixed Interval Hypergraphs

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 [2,3]: 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 χ̄(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 αA(H) is the maximum cardinality of a subset of vertices which contains no co-edge. A mixed hypergraph H is called co-perfect if χ̄(H ′) = αA(H ′) for every subhypergraph H ′. Such minimal non co-perfect hypergraphs as monostars and cycloids C 2r−1 have been found in [3]. A new class of non co-perfect mixed hypergraphs called covered co-bistars is found in this paper. It is shown that mixed interval hypergraphs are co-perfect 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. 1. Basic notions. The following problem was described in [3]: ”Let X = {x1, x2, . . . , xn} be a set of sources of power supply such that the action time of any source is one quantum of time and all sources acting for any given quantum of time switch on and switch off synchronously. Consider the following general constraints on their common work: 1) let A = {A1, A2, . . . , Ak}, Ai ⊆ X, i = 1, . . . , k, k ≥ 1, be a family of subsets of X such that at least two sources from every Ai act for the same quantum of time; 2) let E = {E1, E2, . . . , Em}, Ej ⊆ X, j = 1, . . . ,m, m ≥ 1, be a family of subsets of X such that at least two sources from every Ej act for different quanta of time. ∗Discrete Applied Mathematics, 77 (1997) 29-41; the second author was partially supported by VW–Stiftung Project No. I/69041