An approximate König's theorem for edge-coloring weighted bipartite graphs

Abstract

We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in [0,1] and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König's theorem says that the number of colors needed is exactly the maximum degree. For this generalization, we show that 2. 557 n + o(n) colors are sufficient where n is the maximum total weight adjacent to any vertex, improving the previously best bound of 2. 833n+O(1) due to Du et al. This question is motivated by the question of the rearrangeability of 3-stage Clos networks. In that context, the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unit-sized bins needed to pack the weights of the incident edges. In that setting, we are able to improve the bound to 2. 5480 n + o(n), also improving a bound of 2. 5625n+O(1) of Du et al. Our analysis is interesting in its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one-dimensional bin packing instance which we can prove allows perfect packing.