Decomposing complete equipartite graphs into odd square‐length cycles: number of parts odd

In this article, we introduce a new technique for obtaining cycledecompositions 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. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:401‐414, 2010