Edge-coloring algorithms

The edge-coloring problem is one of the fundamental problems on graphs, which often appears in various scheduling problems like the file transfer problem on computer networks. In this paper, we survey recent advances and results on the classical edge-coloring problem as well as the generalized edge-coloring problems, called the f-coloring and fg-coloring problems. In particular we review various

upper bounds on the minimum number of colors required to edge-color graphs, and present efficient algorithms to edge-color graphs with a number of colors not exceeding the upper bounds. 1 A survey of the edge-coloring problem 1.1 A history of the edge-coloring problem The edge-coloring problem is one of the fundamental problems on graphs. A graph G = (V, E) is an ordered pair of vertex set V and edge set E. An edge in E joins two vertices in V. Throughout the paper we let n = IV[ and m = [E I. The edge-coloring problem is to color all edges of a given graph with the minimum number of colors so that no two adjacent edges are assigned the same color. Fig. I illustrates an edge-coloring of a graph with four colors. A set of edges which are not adjacent each other is called a matching. Since each set of edges colored with the same color is a matching, an edge-coloring of a graph is indeed a partition of E to matchings. We now historically review the edge-coloring problem. The edge-coloring problem was posed in 1880 in relation with the well-known four-color conjecture: every map could be colored...

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