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\(C_{p}\)-decompositions of some regular graphs. (English) Zbl 1087.05048

Let \(C_n\) denote the cycle of length \(n\). A \(C_n\)-decomposition of a graph \(X\) is a partition of its edge set so that each part is isomorphic to \(C_n\). A \(C_n\)-factorization of a graph \(X\) is a partition of its edge set into 2-factors \(F_1,F_2,\dots,F_d\) such that each \(F_i\) is a vertex-disjoint union of cycles \(C_n\). Let \(K_{m(n)}\) denote the complete multipartite graph with \(m\) parts each having cardinality \(n\). A long-standing conjecture is that the obvious arithmetical conditions for \(K_{m(n)}\) to admit a \(C_t\)-decomposition are, in fact, sufficient. This paper essentially adresses the conjecture and establishes several nice results. The most striking result is that \(K_{m(n)}\) admits a \(C_p\)-decomposition, where \(p\) is a prime bigger than 10, if and only if \(n(m-1)\) is even and \(p\) divides \(m(m-1)n^2\).

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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[1] Alspach, B.; Gavlas, H., Cycle decompositions of \(K_n\) and \(K_n - I\), J. Combin. Theory Ser. B, 81, 77-99 (2001) · Zbl 1023.05112
[2] Alspach, B.; Schellenberg, P. J.; Stinson, D. R.; Wagner, D., The Oberwolfach problem and factors of uniform odd length cycles, J. Combin. Theory Ser. A, 52, 20-43 (1989) · Zbl 0684.05035
[3] Billington, E. J., Decomposing complete tripartite graphs into cycles of length 3 and 4, Discrete Math., 197/198, 123-135 (1999) · Zbl 0930.05072
[4] Billington, E. J.; Hoffman, D. G.; Maenhaut, B. M., Group divisible Pentagon systems, Utilitas Math., 55, 211-219 (1999) · Zbl 0938.05020
[5] Cavenagh, N. J., Decompositions of complete tripartite graphs into \(k\)-cycles, Austral. J. Combin., 18, 193-200 (1998) · Zbl 0924.05051
[6] Cavenagh, N. J.; Billington, E. J., Decompositions of complete multipartite graphs into cycles of even length, Graphs Combin., 16, 49-65 (2000) · Zbl 0944.05083
[7] Cavenagh, N. J.; Billington, E. J., On decomposing complete tripartite graphs into 5-cycles, Austral. J. Combin., 22, 41-62 (2000) · Zbl 0965.05078
[8] Lindner, C. C.; Rodger, C. A., Decomposition into cycles II: cycle systems, (Dinitz, J. H.; Stinson, D. R., Contemporary Design Theory, A Collection of Surveys (1992), Wiley-Interscience: Wiley-Interscience New York), 325-369 · Zbl 0774.05078
[9] Lindner, C. C.; Rodger, C. A., Design Theory (1997), CRC Press: CRC Press New York · Zbl 0926.68090
[10] Liu, J., The equipartite Oberwolfach problem with uniform tables, J. Combin. Theory Ser. A, 101, 20-34 (2003) · Zbl 1015.05074
[11] E.S. Mahmoodian, M. Mirzakhani, Decomposition of complete tripartite graphs into 5-cycles, in: C.J. Colbourn, E.S. Mahmoodian (Eds.), Combinatorics and Advances, 1995.; E.S. Mahmoodian, M. Mirzakhani, Decomposition of complete tripartite graphs into 5-cycles, in: C.J. Colbourn, E.S. Mahmoodian (Eds.), Combinatorics and Advances, 1995. · Zbl 0837.05088
[12] R.S. Manikandan, P. Paulraja, \( C_5\); R.S. Manikandan, P. Paulraja, \( C_5\) · Zbl 1120.05071
[13] R.S. Manikandan, P. Paulraja, \( C_7\); R.S. Manikandan, P. Paulraja, \( C_7\) · Zbl 1366.05081
[14] Muthusamy, A.; Paulraja, P., Factorizations of product graphs into cycles of uniform length, Graphs Combin., 11, 69-90 (1995) · Zbl 0822.05054
[15] Piotrowski, W. L., The solution of the bipartite analogue of the Oberwolfach problem, Discrete Math., 97, 339-356 (1991) · Zbl 0765.05080
[16] Šajna, M., Cycle decompositions III: complete graphs and fixed length cycles, J. Combin. Designs, 10, 27-78 (2002) · Zbl 1033.05078
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