## $$C_{p}$$-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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### References:

 [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, (), 325-369 · Zbl 0774.05078 [9] Lindner, C.C.; Rodger, C.A., Design theory, (1997), 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. · Zbl 0837.05088 [12] R.S. Manikandan, P. Paulraja, $$C_5$$-decompositions of some regular graphs, submitted for publication. · Zbl 1120.05071 [13] R.S. Manikandan, P. Paulraja, $$C_7$$-decompositions of some regular graphs, submitted for publication. · 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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