Some Topics On The Coloring Theory Of Cycloids

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring, every C-edgehas at least two vertices of the same color,while every D-edge has at least two vertices colored differently.The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, respectively. Vitaly.Voloshin proposed the conjecture in[2] that the r-uniform hypergraph is co-perfect if and only if it contains neither monostars nor cycloid r≥3,as wholly-edge subhypergraphs. Cycloids play an important role in theories of co-perfect hypergrapgs. We discuss some properties on Cycloids C 2r r?1 .The following is my main results:Theorem 1 Let C 2r r?1 be a cycloid. Thenχ( C 2r r?1 ) = 2r-4.Theorem 2 For any mixed hypergraph H=(X,C,D) ,the following conditions are eqivalent:1)2) H is neither co-bistar nor hole nor a union of two co-bistars without intersecting co-bitransversals ,and H is either a union of three co-bistars without intersecting co–bitransversals,or a union of a co-bistars and a hole without intersecting co-bitransversals,or a bi-hole.Theorem 3 Let C 2r r?1 be a cycloid. For any x∈E, ,and |E|=r, contains co-monostars as wholly-edge subhypergraphs. Theorem 4 Let be a cycloid. If for any x∈E1∪E2, ,and |E1|=|E2|=r , then contains co-monostars as wholly-edge subhypergraphs.Theorem 5 For any contains co-monostars as wholly-edge subhypergraphs.Corollary 6 For any families E of edges of K 2r r?1 , E satisfies ,for any Ei,,then -F ontains co-monostars as wholly-edge subhypergraphs.Theorem 7 If is a bihypergraph ,then its chromatic spectrum is gap free,and the feasible set is = {2,…,2r-4 }.