J Austral Math Soc Ser A 58 pp210--218, 1995.

Balanced Directed Cycle Designs Based on Cyclic Groups

Chaufah Nilrat and Cheryl E. Praeger

Abstract

A balanced directed cycle design with parameters (v, k, 1), sometimes called a (v, k, 1)C_k^® -design, is a decomposition of the complete directed graph K_v^® into the edge disjoint directed cycles of length k. A complete classification is given of (v, k, 1)C_k^®-designs admitting the holomorph {F_{a, b} : x ® ax + b | a, b Î Z_v, (a, v) = 1} of the cyclic group Z_v as a group of automorphisms. In particular it is shown that such a design exists if and only if one of (a) k = 2, (b) p º 1(mod k) for each prime p dividing v, or (c) k is the least prime dividing v, k² does not divide v, and p º 1(mod k) for each prime p > k dividing v.

1991 AMS Subject Classification: 05B30, 20B25

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Authors

Chaufah Nilrat

Facualty of Science, Department of Mathematics, Prince of Songkla University, Haad Yai, Thailand.

Cheryl E. Praeger

Department of Mathematics, University of Western Australia, Nedlands WA 6907, Australia.