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Decomposing complete equipartite graphs into cycles of length \(2p\). (English) Zbl 1149.05026

Summary: It is an open problem to determine whether a complete equipartite graph \(K_{m} \ast \bar K_{n}\) (having \(m\) parts of size \(n\)) admits a decomposition into cycles of arbitrary fixed length \(k\) whenever \(m, n\), and \(k\) satisfy the obvious necessary conditions for the existence of such a decomposition. Recently, R. S. Manikandan and P. Paulraja [Discrete Math. 306, No.4, 429–451 (2006; Zbl 1087.05048)] have shown the necessary conditions are indeed sufficient for a decomposition into cycles of length \(p\) where \(p > 5\) is a prime. The case \(p = 3\) was settled by H. Hanani [Discrete Math. 11, 255–369 (1975; Zbl 0361.62067)] and the case \(p = 5\) was settled by E. J. Billington, D. G. Hoffman , and B. M. Maenhaupt [Util. Math. 55, 211–219 (1999; Zbl 0938.05020)]. Here, we extend this result and show that the necessary conditions for the decomposition of \(K_{m} \ast \bar K_{n}\) into cycles of length \(2p\) (where \(p \geq 3\) is a prime) are also sufficient.

MSC:

05C38 Paths and cycles
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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References:

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