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

Abstract

In this paper we show that the complete equipartite graph with n parts, each of size 2k, decomposes into cycles of length λ2 for any even n≥4, any integer k≥3 and any odd λ such that 3≤λ<2nk and λ divides k. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph with an even number of parts into cycles of length p2, where p is prime. In proving our main result, we have also shown the following. Let λ≥3 and n≥4 be odd and even integers, respectively. Then there exists a decomposition of the λ-fold complete equipartite graph with n parts, each of size 2k, into cycles of length λ if and only if λ<2kn. In particular, if we take the complete graph on 2n vertices, remove a 1-factor, then increase the multiplicity of each edge to λ, the resultant graph decomposes into cycles of length λ if and only if λ<2n.