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

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 2 are positive integers with n >= 3, lambda >= 3 and n and lambda both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length lambda(2) whenever nm >= lambda(2) and lambda divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p2, where p is prime. (C) 2010 Wiley Periodicals, Inc. J Combin Designs 18: 401-414, 201