Some Results On Fractional Factor

Factor theory of graph is an important branch in graph theory. It is a subject that receives most research in graph theory study. In our daily life many problems on optimization and network design, e.g., coding design, building blocks, the file transfer problem on computer network, scheduling problem and so on, are related to the factor, factorizations and orthogonal factorizations of graphs [15]. The file transfer problem can be modelled as factor and (0, f)-factorizations(or f-colorings) of a graph. The design of Latin Squares and Room Squares are related to factor and orthogonal factorizations in graph.All graphs considered in this paper will be finite simple graphs. Let G be a graph with vertex set V(G) and edge set E(G), The degree of x in G is denoted by d_G(x).λ(G) andκ(G) denote the edge connectivity and connectivity of G, respectively. S(G) denotes the minimal degree of G. If S is a subset of V(G), G[S] denotes the induced subgraph by S. The set of isolated vertices of G\S is denoted by I(G\S) and |I(G\5)| = i(G\S). For two disjoint subsets S, T of V(G), E_G(S,T) denotes the set of edges whose one vertex in 5 and another in T and |Eg(S,T)| = e_G(S,T).Let g and f be two integer-valued functions such that 0 < g{x) < f(x) for all x∈V(G). A (g,f)-factor F of G is a spanning subgraph of G satisfying g(x)≤d_F{x)≤f(x) for all x∈V(G). A fractional (g, f)-indicator function is a function h that assigns to each edge of graph G a number in [0,1], so that for each vertex x∈V(G) we have g(x)≤d_Gh(x)≤f(x), where d_Gh(x) =∑_e∈Exh(e) is the fractional degree of x∈V(G), with E_x = {e : e = xy∈E(G)}. Let h be a fractional (g, f)-indicator function of a graph G. Set E_h = {e : e∈E(G) and h(e)≠0}. If G_h, is a spanning subgraph of G such that E(G_h) = E_h, then G_h is called a fractional (g,f)-factor of G. If g(x) = f(x) = k(k is a non-negative integer) for all x∈V(G), the a fractional (g.f)-factor is called a fractional k-factor. In particular, a fractional 1-factor is also called a fractional perfect matching.This thesis is divided into six chapters. We mainly study fractional graph from three aspect, and obtain some results.In the paper we study deleted graph, obtained sufficient conditions related to toughness and isolated toughness for a graph to be fractional 1-deleted, 2-deleted and A;-edge-deleted. The results are proved to be best pos- sible in some sense. We have got the sufficient conditions related to toughness and isolated toughness for a graph to be fractional 1-covered, 2-covered are given. The results are proved to be best possible in some sense. In particular, a necessary and sufficient condition for a (g, f)-factor covering a given k-matching is obtained. Lasterly we study fractional k-factor-critical graph and fractional k-extendable graph.