Funding information This work was supported by the Freigeist-Fellowship of the Volkswagen Stiftung; and the German research council (DFG) Research Training Group 2236 UnRAVeL. Abstract The minimum color-degree perfect b-matching problem (Col-BM) is a new extension of the perfect b-matching problem to edgecolored graphs. The objective of Col-BM is to minimize the maximum number of differently colored edges in a perfect b-matching that are incident to the same node. We show that Col-BM is NPhard on bipartite graphs by a reduction from (3,B2)-Sat, and conclude that there exists no (2 − ε)-approximation algorithm unless P = NP. However, we identify a class of two-colored complete bipartite graphs on which we can solve Col-BM in polynomial time. Furthermore, we use dynamic programming to devise polynomialtime algorithms solving Col-BM with a fixed number of colors on series-parallel graphs and simple graphs with bounded tree width.

中文翻译：

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最小色度完美 b 匹配

资金信息 这项工作得到了大众基金会的 Freigeist-Fellowship 的支持；和德国研究委员会 (DFG) 研究培训组 2236 UnRAVeL。摘要 最小颜色度完美 b 匹配问题 (Col-BM) 是完美 b 匹配问题对边彩色图的新扩展。Col-BM 的目标是最小化完美 b 匹配中发生在同一节点上的不同颜色边的最大数量。我们通过从 (3,B2)-Sat 减少来证明 Col-BM 在二部图上是 NPhard，并得出结论，除非 P = NP，否则不存在 (2 − ε) 近似算法。然而，我们确定了一类可以在多项式时间内求解 Col-BM 的双色完全二分图。此外，