On Edge-coloring Problems Of Graphs

Throughout this paper, a graph G(V, E) allows multiple edges but no loops and has a finite vertex set V(G) and a finite and nonempty edge set E(G). dc(v) is the degree of the vetex v of G for all v 6 V. Let A(G) and 8(G) denote the maximum degree and the minimum degree of G, respectively. Given a graph G, let f₁ and f₂ be two integer-valued functions defined on V(G) such that 0 ≤ f₁(v) ≤ f₂(v) ≤ d(v) for all v ∈ V. If there exists a coloring of E(G) in which every edge set with the same color is a spanning subgraph H of G satisfyingf₁(v)≤d_H(v)≤f₂(v),(?)v∈V(G),then the edge coloring of G is called an (f₁,f₂)-edge coloring of G. We call the edge set with the same color an (f₁,f₂)-factor. So an (f₁,f₂)-edge coloring of G with k colors divides the edge set E(G) into edge disjoint (f₁,f₂)-factors F₁, F₂,…, F_k, it is also called an (f₁,f₂)-factorization.If f₁(v) = 0 and f₂(v) = 1 for each vertex v of G, then an (f₁,f₂)-edge coloring of G is an ordinary (proper) edge coloring. The minimum number of colors needed to properly edge coloring G is called the chromatic index of G, and is denoted by x(G). Vizing proved that for any multigraph G,holds. For simple graph G, the chromatic index x'(G) is either â–³(G) or â–³(G) - 1. A simple graph G is said to be of class one if x'(G) = â–³(G), and of class two if x'(G) = â–³(G) + 1. The research on the classification of simple graphs is one important branch of graph theory.If f₁(v) = 0 and f₂(v) = f(v) for all vertex v ∈ V, an (f₁,f₂)-edge coloring of G is an f-edge coloring. The minimum number of colors needed to f-edge coloring G is called the /-chromatic index of G, and is denoted by x_f'(G)- Given a positive integer-valued function g(u, v) for a pair u and v of distinct vertices, if there exitsan /-edge coloring C such that the number of edges of G with the same color and joining u and v is not greater than g(u, v), then such an /-edge coloring is called an /9-edge coloring of G. The minimum number of colors needed to /9-edge coloring G is called the /^-chromatic index of G, and is denoted by Xfg{G)- The /-edge coloring is a generalization of the ordinary edge coloring.If /,(?) = l and f2(v) = d{v) for all vertex v € V, an (/i.^J-edge coloring of G is an ordinary edge cover coloring. The maximum number of colors needed to edge cover coloring G is called the edge cover chromatic index of G, and is denoted by x'c{G)- Gupta proved that any multigraph G has6(G)-n(G)?$l(d(v)-n{v))/f(v)\.Chapter Three is major in g-super edge cover-coloring. In this chapter we have the following result.Theorem 3.3.1 Let G(V,E) be a graph and 1 < f{v) < [(d(v) - fi{v))/n\ for all v € V and 1 < g(vw) < min{f(v), f{w)}, for all vw e E. Thenx'fg(G) > mm [(d(v) - n(vw)/g(vw))/f(v)\.Futhermore, In Chapter Two and Chapter Three, we list some problems for future research.