Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees

Abstract

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E = { E 1 , … , E m } , together with integers s i and t i ( 1 ≤ s i ≤ t i ≤ | E i | ) for i = 1 , … , m . A vertex coloring φ is feasible if the number of colors occurring in edge E i satisfies s i ≤ | φ ( E i ) | ≤ t i , for every i ≤ m . In this paper we point out that hypertrees–hypergraphs admitting a representation over a (graph) tree where each hyperedge E i induces a subtree of the underlying tree–play a central role concerning the set of possible numbers of colors that can occur in feasible colorings. We also consider interval hypergraphs and circular hypergraphs, where the underlying graph is a path or a cycle, respectively. Sufficient conditions are given for a ‘gap-free’ chromatic spectrum; i.e., when each number of colors is feasible between minimum and maximum. The algorithmic complexity of colorability is studied, too. Compared with the ‘mixed hypergraphs’–where ‘D-edge’ means ( s i , t i ) = ( 2 , | E i | ) , while ‘C-edge’ assumes ( s i , t i ) = ( 1 , | E i | − 1 ) –the differences are rather significant.