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The problem of decomposing a complete tripartite graph into 5-cycles wasfirst proposed in 1995 by Mahmoodian and Mirzakhani and since then manyattempts have been made to decompose such graphs into 5-cycles. Such attemptswere partially successful but parts of the problem still remain open. In thispaper, we investigate the problem deeper, decompose more tripartite graphs into5-cycles, and introduce the Graph Adjacency Matrix (GAM) method for cycledecomposition in general. GAM method transforms the cycle decomposition problemto covering squares with certain polygons. This new formulation is easier tosolve and enables us to find explicit decompositions for numerous cases thatwere not decomposed before.

Decomposition of complete tripartite graphs into 5-cycles using the graph adjacency matrix