Integer programming models for colorings of mixed hypergraphs

A mixed hypergraph H = (X, C,D) consists of the vertex set X and two families of subsets: the family C of C-edges and the family D of D-edges. In a coloring, every C-edge has at least two vertices of common color, while every D-edge has at least two vertices of different colors. The largest (smallest) number of colors for which a coloring of a mixed hypergraph H using all the colors exists is called the upper (lower) chromatic number and is denoted χ̄(H) ( χ(H) ). We consider integer programming models for colorings of mixed hypergraphs in order to show that algorithms for optimal colorings may be transformed and used for finding optimal solutions of the respective integer programming problems. 1 Mixed hypergraphs A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, and each of C, D is a family of subsets of X, called C-edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a mapping from X into a set of k colors so that each C-edge has at least two vertices of a common color and each D-edge has at least two vertices of different colors. That means that in every coloring no C-edge is polychromatic (i.e. no C-edge has all the colors different) and no D-edge is monochromatic. A mixed hypergraph is k-colorable if it has a coloring with at most k colors. If H admits no coloring then it is called uncolorable. A c ©2000 by D.Lozovanu, V.Voloshin 64 Integer programming models for . . . strict k-coloring of a mixed hypergraph is a proper coloring using all k colors. The minimum number of colors in a coloring of H is its lower chromatic number χ(H); the maximum number of colors in a strict coloring is its upper chromatic number χ̄(H). Classical coloring theory of hypergraphs with edge set E [1] is the special case where the family of C-edges is empty and we color the mixed hypergraph (X, ∅, E). Coloring of mixed hypergraphs is a new topic introduced in [3, 4]. 2 Integer programming models There are several ways to formulate the colorability problem for mixed hypergraphs as an integer programming problem. Consider H = (X, C,D), where X = {x1, x2, . . . , xn}, n ≥ 1, C = {C1, C2, . . . , Cl}, l ≥ 1, and D = {D1, D2, . . . , Dm}, m ≥ 1. Let us have n colors. and A = (αij) be a (0,1)-matrix allocating vertices to colors in such a way that αij = { 1 if vertex xi is colored with colorj, 0 otherwise. Then the condition that each vertex receives exactly one color is: