Decomposing complete equipartite graphs into odd square-length cycles: Number of parts odd

摘要

In this article, we introduce a new technique for obtaining cycle decompositions of complete equipartite graphs from cycle decompositions of related multigraphs. We use this technique to prove that if n, m and λ are positive integers with n ≥ 3, λ≥ 3 and n and λ both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length λ~2 whenever nm ≥ λ~2 and λ divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p~2, where p is prime.