Decomposing complete tripartite graphs into 5-cycles when the partite sets have similar size
\textit{E. S. Mahmoodian} and \textit{M. Mirzakhani} [Papers presented at the 25th annual Iranian mathematics conference, AIMC 25, held at Sharif University of Technology in Tehran, Iran, March 28--31, 1994. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 329, 235--241 (1995; Zbl 0837.05088)] showed necessary conditions for a complete tripartite graph to be decomposed into 5-cycles. Many cases of the problem of finding necessary and sufficient conditions a complete tripartite graph to be decomposed into 5-cycles have been solved, however, the case when the partite sets have odd and distinct sizes remains open. The authors introduce a new approach by embedding previously known decompositions into larger ones. It is shown that when the partite sets are of asymptotically similar sizes, the conjectured necessary conditions for a decomposition are also sufficient. Specifically, it is shown that if \(R\), \(S\) and \(T\) are odd integers such that \(RS+ST+TR\) is divisible by 5 and \(100 < R < S < T < \kappa R\), where \(\kappa = -\frac{95}{16}+\frac{3}{16}\sqrt{1401} = 1.0806\dots\), then the complete tripartite graph having three parts of sizes \(R\), \(S\) and \(T\), has a decomposition into 5-cycles.
scientific article; zbMATH DE number 5996234