# The generalized almost resolvable cycle system problem

## Abstract

Let (X, C) be a k-cycle system of order n, with vertex set X (of cardinality n) and collection of k-cycles C. Suppose n=kq+r where r<k. An almost parallel class of C is a collection of q=(n−r)/k pairwise vertex-disjoint k-cycles of C. Each almost parallel class thus will miss r of the n vertices in X. The k-cycle system (X,C) is said to be almost resolvable if C can be partitioned into almost parallel classes such that the remaining k-cycles are vertex disjoint. (These remaining k-cycles are referred to as a short parallel class.)

In this paper we (a) construct an almost resolvable 10-cycle system of every order congruent to 5 (mod 20), and (b) construct an almost resolvable 10-cycle system of order 41, thus completing the result that the spectrum for almost resolvable 10-cycle systems consists of all orders congruent to 1 or 5 (mod 20).

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## References

1. [1]

B. Alspach and H. Gavlas: Cycle decompositions of K n and K nI, J. Combin. Theory Ser. B 81 (2001), 77–99.

2. [2]

B. Alspach, P. J. Schellenberg, D. R. Stinson and D. Wagner: The Oberwolfach problem and factors of uniform odd length cycles, J. Combin. Theory Ser. A 52 (1989), 20–43.

3. [3]

I. J. Dejter, C. C. Lindner, M. Meszka and C. A. Rodger: Almost resolvable 4-cycle systems, J. Combin. Math. Combin. Computing 63 (2007), 173–182. Corrigendum/Addendum ibid. 66 (2008), 297–298.

4. [4]

C. C. Lindner, M. Meszka and A. Rosa: Almost resolvable cycle systems — an analogue of Hanani triple systems, J. Combin. Designs 17(5) (2009), 404–410.

5. [5]

C. C. Lindner and C. A. Rodger: Design Theory, CRC Press (1997), 208 pp.

6. [6]

W. L. Piotrowski: The solution of the bipartite analogue of the Oberwolfach problem, Discrete Math. 97 (1991), 339–356.

7. [7]

M. Šajna: Cycle decompsoitions III: complete graphs and fixed length cycles; J. Combin. Designs 10 (2002), 27–78.

8. [8]

S. A. Vanstone, D. R. Stinson, P. J. Schellenberg, A. Rosa, R. Rees, C. J. Colbourn, M. Carter and J. Carter: Hanani triple systems, Israel J. Math. 83 (1993), 305–319.

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