Color-bounded hypergraphs, I: General results

Abstract

The concept of color-bounded hypergraph is introduced here. It is a hypergraph (set system) with vertex set X and edge set E = { E 1 , … , E m } , where each edge E i is associated with two integers s i and t i such that 1 ≤ s i ≤ t i ≤ | E i | . A vertex coloring φ : X → N is considered to be feasible if the number of colors occurring in E i satisfies s i ≤ | φ ( E i ) | ≤ t i , for all i ≤ m . Color-bounded hypergraphs generalize the concept of ‘mixed hypergraphs’ introduced by Voloshin [V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45–52], and a recent model studied by Drgas-Burchardt and Łazuka [E. Drgas-Burchardt, E. Łazuka, On chromatic polynomials of hypergraphs, Applied Mathematics Letters 20 (12) (2007) 1250–1254] where only lower bounds s i were considered. We discuss the similarities and differences between our general model and the more particular earlier ones. An important issue is the chromatic spectrum–strongly related to the chromatic polynomial–which is the sequence whose k th element is the number of allowed colorings with precisely k colors (disregarding color permutations). Problems concerning algorithmic complexity are also considered.