Circular mixed hypergraphs II: The upper chromatic number

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 (Cor 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 umber χ̄ and sieve number s, then n− s− 2 ≤ χ̄ ≤ n− s + 2.