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, Konig'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 improvinga 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.
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机译:我们考虑边缘着色二部图的一般化,其中每个边缘的权重为[0,1],并且边缘的着色必须满足入射到顶点 v I的边缘的权重之和。 >任何颜色的最大数量必须为1。对于单位权重,柯尼希定理说,所需的颜色数量恰好是最大程度。对于这种概括,我们显示出2. 557个 n I> + O I>( n I>)颜色,其中 n I>为Du等人指出,与任何顶点相邻的最大总重量提高了以前的最佳界限2。833 n I> + O I>(1)。这个问题是由三级Clos网络的可重排性问题引起的。在这种情况下,边缘着色问题中感兴趣的相应参数 n I>是包装入射边缘的权重所需的单位大小仓位数量的所有顶点上的最大值。在这种情况下,我们可以将绑定范围提高到2。5480 n I> + o I>( n I>),也可以将绑定范围提高到2。5625 Du等人的 n I> + O I>(1)。我们的分析本身很有趣,涉及二部图的新颖分解结果,并引入了相关的连续一维箱式装箱实例,我们可以证明该实例可以实现完美装箱。
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