Circular Mixed Hypergraphs III: C–perfection

A mixed hypergraph is a tripleH = (X, C,D), whereX 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 an injective mapping c : X → {1, . . . , k} such that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. Maximum number of colors in a coloring using all the colors is called upper chromatic number χ̄(H). Maximum cardinality of subset of vertices which contains no C-edge is C-stability number αC(H). A mixed hypergraph is called C-perfect if χ̄(H′) = αC(H) for any induced subhypergraph H′. 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 on the host cycle. We investigate the problem of C-perfection of circular mixed hypergraphs.