Pattern hypergraphs

Abstract

The notion of pattern hypergraph provides a unified view of several previously studied coloring concepts. A pattern hypergraph H is a hypergraph where each edge is assigned a type pPi that determines which of possible colorings of the edge are proper. A vertex coloring of H is proper if it is proper for every edge. In general, the set of integers k such that H can be properly colored with exactly k colors need not be an interval. We find a simple sufficient and necessary condition on the edge types pP1,...,Pλ for the existence of a pattern hypergraph H with edges of types P1,..., Pλ such that the numbers of colors in proper colorings of H do not form an interval of integers.