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For a d-regular bipartite multigraph, an edge coloring is equivalent to a decomposition of the edge set into d perfect matchings. When d is a power of 2, we can recursively perform Euler partitions to find the perfect matchings. When d is not a power of 2, however, we eventually reach a subproblem graph of odd degree where we can no longer perform an Euler partition. We propose two different algorithms that address this case. Both algorithms make use of an auxiliary matching, called dummy edges, to make the degree of the graph even. In the first algorithm, dummy edges are added prior to tracing out cycles. In the second algorithm, we add dummy edges as a way to close paths that have already been traced. We will analyze both algorithms separately and also consider a hybrid version.
Senior Honors Thesis. Advisor: Thomas Cormen.
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Patricia T. Neckowicz, "Two Algorithms for Finding Edge Colorings in Regular Bipartite Multigraphs." Dartmouth Computer Science Technical Report TR2015-771, May 2015.
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