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Balanced directed cycle designs based on cyclic groups

  • Chaufah Nilrat (a1) and Cheryl E. Praeger (a2)

Abstract

A balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

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Copyright

References

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[1]Bermond, J. C. and Sotteau, D., ‘Graph decompositions and G-designs’, Proceedings of the Fifth British Combinatorial Conference, Cong. Numer. 15 (1975), 5372.
[2]Brand, N. and Huffman, W. C., ‘Mendelsohn designs admitting the affine group’, Geom. Dedicata 22 (1987), 173196.
[3]Hell, P. and Rosa, A., ‘Graph decompositions, handcuffed prisoners and balanced P-designs’, Discrete Math. 2 (1972), 229252.
[4]Praeger, C. E., ‘Balanced directed cycle designs based on groups’, Discrete Math. 92 (1991), 275290.
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Balanced directed cycle designs based on cyclic groups

  • Chaufah Nilrat (a1) and Cheryl E. Praeger (a2)

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