Applications Of Probabilistic Methods In Combinatorics And Coloring Theory Of Mixed-Hypergraph

The probabilistic method is a powerful tool for dealing with many problems in combinatorics and number theory. This method is roughly divided into two types:one is constructive method, the other is non-constructive method. It used to determine the existence of a combinatorics object with specific properties. This paper will use such non-constructive probabilistic method to solve the problems in coloring theory of mixed-hypergraphs.A mixed-hypergraph consists of two families of edges:the C-edges and D-edges. The main difference between them is coloring requirements:every D-edge at least has two vertices colored differently, while every C-edge at least has two vertices of the same color. Discussing mixed hypergraphs in special typies is a good method, and finding chromatic spectrum and chromatic feasible set of it is a well direction in researching coloring theory of the mixed hypergraphys.In this paper, first we research coloring problems of the set of natural num-bers using non-constructive probabilistic methods. Then deal with coloring prob-lems of several special types of mixed-hypergraphs, upper chromatic number of C-hypergraphys, lower chromatic number of D-hypergraphys and chromatic num-ber of mixed hypergraphys of several typies. Finally, we give the algorithm of finding chromatic feasible set of several typies mixed hypergraphs.