Circular mixed hypergraphs II:The upper chromatic number

Abstract

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, the C -edges and D -edges, respectively. A proper k-coloring of H is a mapping c : X → [ k ] such that each C -edge has two vertices with a common color and each D -edge has two vertices with distinct colors. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge ( C - or D -) induces a connected subgraph of this cycle. We suggest a general procedure for coloring circular mixed hypergraphs and prove that if H is a reduced colorable circular mixed hypergraph with n vertices, upper chromatic number χ ¯ and sieve number s , then n - s - 2 ⩽ χ ¯ ⩽ n - s + 2 .