(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 8, No. 3, 2017 250 | Page www.ijacsa.thesai.org A Low Complexity based Edge Color Matching Algorithm for Regular Bipartite Multigraph Rezaul Karim Dept. of Computer Science & Engineering University of Chittagong (CU) Chittagong, Bangladesh Muhammad Mahbub Hasan Rony Dept. of Computer Science & Engineering International Islamic University Chittagong (IIUC) Chittagong, Bangladesh Md. Rashedul Islam Dept. of Computer Science & Engineering International Islamic University Chittagong (IIUC) Chittagong, Bangladesh Md. Khaliluzzaman* Dept. of Computer Science & Engineering International Islamic University Chittagong (IIUC) Chittagong, Bangladesh 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. Keywords—matching; edge-coloring; complexity; bipartite multigraph; DFS I. INTRODUCTION An edge coloring of a Graph is one of the well-known, exoteric researched topics in the arena of graph theory. Edge coloring of a graph G is used various colors, so that, the adjacent edges are obtained different colors. By using this concept of edge coloring many real-world problems can be solved. An edge coloring has applications in scheduling problems and in frequency assignment for fiber optic networks. It also used to solve the timetabling problem, register allocation, pattern matching, designing seating plans, solving Sudoku puzzles and so on. This section provides 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. For example, in [1], proposed a method for edge coloring in which every (3, Δ)-bipartite graph G, chromatic index ≤ 4Δ. This paper only considered the (3, Δ)-bipartite simple graph. In [2], proposed an edge coloring method for course timetabling. One-sided interval colorings of a bipartite graph method are introduced in [3]. For any graph G with bipartite set (X, Y) where authors present upper    (G, X) for classes of bipartite graphs G with maximum degree ∆ (G) at most 9. In particular, if ∆ (G) = 4, then    (G, X) ≤ 6 and so on. In [4], authors derived a theorem to find the closed paths from C =MN for matching M and N. A closed path C can be found in (|C|) time on average. This theorem helped to develop the proposed algorithm for minimal edge-coloring. This concept of open and closed path can be easily obtained from this theorem. In [5], showed a theorem in which any edge color matching of a complete bipartite graph K n,n contains 18 vertexes with three colors. This method creates disjoint monochromatic cycles which together cover all vertices. The minimum number of cycles is required for this type of covering is 5. In [6], proposed an algorithm to find out two disjoint matching M 1 and M 2 for a given (X, Y) bipartite graph with set SX, where, M 1 saturates X and M 2 saturates S. The problem was solved by finding and appropriate factor of the graph when |S|≥|X|-1. In [7], proved a method for two bipartite graphs G and H, where, H is a fixed graph whose vertices will be shown as colors. And H-coloring of a graph G is a process of assigning colors for preserving adjacency in graph G. In this paper, 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. For that, the proposed method is divided into some parts that are run with an independent time complexity and helps to reduce the overall time complexity. The edge coloring of a bipartite multigraph is highly related to finding a perfect matching efficiently. To obtain the perfect edge color matching the proposed method is divided into two parts. In the first part, the Depth First Search (DFS) algorithm is used to extract the closed and opened path circuits as well as bad edges. In the second part, the bad edges are rearranged to find the perfect edge color matching. The rest of the paper is organized as follows. The preliminaries of the graph theory are described in Section II. The proposed minimal edge color matching algorithm is introduced in the next section. Case studies are described in Section IV. Experimental results and discussions are explained in Section V. The papers are concluded in Section VI.