This paper discusses uniquely fractional colourable graphs. Suppose r=n/k is a rational. Two definitions of uniquely r-fractional colourable graphs are given, and we prove that the definitions are equivalent. Then we prove that for any rational n/K≥2, for any integer g, there exists a uniquely n/k -fractional colourable graph G of girth at least g, and of maximum degree at most 5n13k.

Let G = (X, Y,E(G)) be a bipartite graph with vertex set V (G) = X [ Y and edge set E(G) and let g and f be two non-negative integer-valued functions defined on V (G) such that g(x)≤f(x) for each x ∈ V (G). A (g, f)-factor of G is a spanning subgraph F of G such that g(x)≤dF (x)≤f(x) for each x ∈ V (F); a (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. In this paper it is proved that every bipartite (mg+ m− 1,mf− m+ 1)-graph has (g, f)-factorizations randomly k-orthogonal to any given subgraph with km edges if k ≤ g(x) for any x ∈ V (G) and has a (g, f)-factorization k-orthogonal to any given subgraph with km edges if k-1≤g(x) for any x ∈ V (G) and that every bipartite (mg,mf)-graph has a (g, f)-factorization orthogonal to any given m-star if 0≤g(x)≤f(x) for any x ∈ V (G). Furthermore, it is shown that there are polynomial algorithms for finding the desired factorizations and the results in this paper are best possible.

Let G be a graph with vertex set V (G). Let n, k and d be non-negative integers such that n+2k+d ≤ |V (G)|−2 and |V (G)|−n−d is even. A matching which covers exactly |V (G)|−d vertices of G is called a defect-d matching of G. If when deleting any nvertices of G the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is called a (n, k, d)−graph. In this paper a characterization of (n, k, d)-graphs is given and several properties (such as connectivity, minimum degree, hierarchy, etc.) of (n, k, d)-graphs are investigated.

G（V,E）是至少含有一条边的无环图，f是定义在V上的整值函数且对任意的v∈V，有1≤f(v)≤d(v)。若边染色G使所用的每一种颜色在任一顶点v上至少出现f(v)次，则称该滚氛C为f-边覆盖染色。能对图G进行f-边覆盖k-边染色的最大颜色数k，称为图G的f-边覆盖色数，记为Xfc（G）。本文提供了一个关于xfc（G）的Vizing型定理，使一些已有重要结论得以推广；研究了一些使Xfc（G）达到该Vizing型定理上界的几类图或函数f；最后，还讨论了f-边盖染色的变型，提出了一些可进一步研究的问题。

Let G be a bipartite graph with vertex set V (G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V (G) such that g(x) ≤ f(x) for every vertex x of V (G). A (g, f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g, f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.

The upper chromatic number x (H) of a C-hypergraph H = (X, C) is the maximum number of colors that can be assigned to the vertices of H in such a way that each C ∈ C contains a monochromatic pair of vertices. It is closely related to the number of C-edges. This paper discusses the relationship between the lower bound of the size of C-edges and the lower bound of the upper chromatic number and gives an upper bound of minimum C-edge number of 4-uniform C-hypergraphs with minimum upper chromatic number.

An edge-colouring of a graph G is equitable if, for each vertex v of G, the number of edges of any one colour incident with v di ers from the number of edges of any other colour incident with v by at most one. Hilton and de Werra have proved that if k does not divide d(v) for all vertex v ∈ V (G), then G has an equitable edge-colouring with k colours. In this paper, using the result of Hilton and de Werra, we give the very simple proofs of several results on f-edge colourings and f-edge cover colourings the original proofs of which are diffcult. Furthermore, we give some new sharp su cient conditions for a graph to have (g, f)-factorizations which improve the results in [8].

G) and edge set E(G) and let g and f be two integervalued functions defined on V (G) such that 2k−1≤g(x)≤f(x) for all x∈V (G). Let H be a subgraph of G with mk edges. In this paper it is proved that every (mg+m−1,mf−m+1)-graph G has (g, f)-factorizations randomly k-orthogonal to H and shown that the result is best possible.

A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple suffcient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b.