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

TL;DR

This paper advances the understanding of decomposing complete tripartite graphs into 5-cycles by introducing the Graph Adjacency Matrix method, which simplifies the problem and enables explicit solutions for previously unresolved cases.

Contribution

It introduces the GAM method for cycle decomposition, transforming the problem into covering squares with polygons, and extends decomposition results for tripartite graphs.

Findings

Successfully decomposed more tripartite graphs into 5-cycles

Introduced the GAM method for cycle decomposition

Provided explicit decompositions for new cases

Abstract

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.