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