Generalized Edge Coloring for Channel Assignment in Wireless Networks

This paper introduces a new graph theory problem called generalized edge coloring (g.e.c). A generalized edge coloring is similar to traditional edge coloring, with the difference that a vertex can be adjacent to up to k edges that share the same color. The concept of generalized edge coloring can be used to formulate the channel assignment problem in multi-channel multi-interface wireless networks. We provide theoretical analysis for this problem. Our theoretical findings can be useful for system developers of wireless networks. We show that when k = 3, there are graphs that do not have generalized edge coloring that could achieve the minimum number of colors for every vertex. On the contrary, when k = 2 we show that if we are given one extra color, we can find a generalized edge coloring that uses the minimum number of colors for each vertex. In addition, we show that for certain classes of graphs we are able to find a generalized edge coloring that uses the minimum number of colors for every vertex without the extra color. These special classes of graphs include bipartite graph, graphs with a power of 2 maximum degree, or graphs with maximum degree no more than 4