Abstract
An edge coloring of a graph G is a process of assigning colors to the adjacent edges so that the adjacent edges represents the different colors. In this paper, an algorithm is proposed to find the perfect color matching of the regular bipartite multigraph with low time complexity. For that, the proposed algorithm is divided into two procedures. In the first procedure, the possible circuits and bad edges are extracted from the regular bipartite graph. In the second procedure, the bad edges are rearranged to obtain the perfect color matching. The depth first search (DFS) algorithm is used in this paper for traversing the bipartite vertices to find the closed path, open path, incomplete components, and bad edges. By the proposed algorithm, the proper edge coloring of D – regular bipartite multi-graph can be obtained in O (D.V) time.
Highlights
An edge coloring of a Graph is one of the well-known, exoteric researched topics in the arena of graph theory
We provide a descriptive summary of some methods that have been implemented and tested at graph theory for solving edge coloring problems
An algorithm is proposed that is developed to find a perfect color matching of a regular bipartite multigraph. This is done by dealing edge coloring with lower time complexity
Summary
An edge coloring of a Graph is one of the well-known, exoteric researched topics in the arena of graph theory. We provide a descriptive summary of some methods that have been implemented and tested at graph theory for solving edge coloring problems This topic has gained importance for the purpose of efficient edge color matching in the different graphs. A closed path C can be found in (|C|) time on average This theorem helped to develop the proposed algorithm for minimal edge-coloring. In [5], showed a theorem in which any edge color matching of a complete bipartite graph Kn,n contains 18 vertexes with three colors This method creates disjoint monochromatic cycles which together cover all vertices. An algorithm is proposed that is developed to find a perfect color matching of a regular bipartite multigraph This is done by dealing edge coloring with lower time complexity.
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