The Minimum Vertices Number Of R-uniform Bi-hypergraphed Zhu Xiao

The coloring theory of the mixed hypergraphs is a dual and generalize of the classical coloring theory of graph and hypergraph. But there are a lot of different natur about the coloring theory between mixed hypergraphs and the classical hy-pergraph. For example, there may indeed be gaps in the chromatic spectrum of mixed hypergraphs. It is the essential difference between mixed hypergraph and the classical hypergraph and is important for the coloring theory of mixed hypergraph and its application. Hence, one of the important research directions is the research of the smallest mixed hypergraph which is a realization of a given feasible set S. Note that, the minimum numbers of vertices and edges in a mixed hypergraph whose feasible set is S are all important.Based on the extremum problems of3-uniform bi-hypergraph which is a one-realization of S, this paper reconstructed the hypergraph and discussed about3-uniform bi-hypergraph which is a realization of S. The research methods is to increasing the dimensions of the corresponding component of the spectrum or re-moving some edges which constraint the corresponding component of the spectrum. This is the first key issue of this paper.Note that, the upper chromatic number of mixed hypergraph is decided by C-hypergraph. On the basis of function f(n,3) which is equal to the minimum edge number of C-hypergraphs with minimum upper chromatic number, we give the related conclusion about bi-hypergraphs. This is the second key issue of this paper. And this paper is organized as follows:In the first chapter, we first put forward coloring problem of mixed hypergraph, and then further discuss its application, basic definition and abundant theoretical content. And we point out the main study frame of this paper in the last section.In the second chapter, we discuss some basic conceptions, theorems and prop-erties involving this paper about the feasible set of mixed hypergraph, and analyze the main results on the extreme value problem of the realization of the feasible set achieved by forefathers in recent years.In the third chapter, we introduce the main work and the main conclusions of this paper. And we get two results about the minimal realization of3-uniform bi-hypergraph:conclusion1. For integers s≥2, n1> n2>…> ns≥s and t1=0,t2,…,ts≥0, let R2=(r1,r2,…, rn1) be a non-negative vector with rn1=1,rni=2ti,i∈{2,…,s} and rj=0,j∈[n1]\{n1,n2,…,ns}. If ni-1-ni> ti,i∈{2,…,s}, then we construct a hypergraph in a multidimensional space and deter-mine that δ3(R2) be the minimum number of vertices of3-uniform bi-hypergraphs which are realizations of R2. As a result, we partially solve an open problem proposed by Kral’in2004.conclusion2. For every n≥3there exists a3uniform bi-hypergraph Hn=(Xn,Bn) with n vertices, upper chromatic number2and [n(n-2)/3] edges. We prove that the minimum edge number of bi-hypergraphs with minimum upper chromatic number.In the fourth chapter, we summarize the main conceptions and properties on the promotion form of mixed hypergraphs. And we provide some research directions which can continue to discuss.