Circular mixed hypergraphs II:The upper chromatic number

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