Abstract
A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph Km(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).
Similar content being viewed by others
References
Alspach, B., Gavlas, H.: Cycle decompositions of K n and K n − I. J. Combin. Theory (Ser. B) 81, 77–99 (2001)
Billington, E.J., Hoffman, D.G.: Equipartite and almost equipartite gregarious 4-cycle systems. Discrete Math. (to appear)
Billington, E.J., Hoffman, D.G., Smith, B.R.: Equipartite gregarious 6- and 8-cycle systems. Discrete Math. 307, 1659–1667 (2007)
Cavenagh, N.J.: Further decompositions of complete tripartite graphs into 5-cycles. Discrete Math. 256, 55–81 (2006)
Cavenagh, N.J., Billington, E.J.: On decomposing complete tripartite graphs into 5-cycles. Australasian J. Combin. 22, 41–62 (2000)
Hanani, H.: Balanced incomplete block designs and related designs. Discrete Math. 11, 255–369 (1975)
Manikandan, R.S., Paulraja, P.: C p -decompositions of some regular graphs. Discrete Math. 306, 429–451 (2006)
Šajna, M.: Cycle decompositions III: Complete graphs and fixed length cycles. J. Combin. Designs 10, 27–78 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Smith, B.R. Equipartite Gregarious 5-Cycle Systems and Other Results. Graphs and Combinatorics 23, 691–711 (2007). https://doi.org/10.1007/s00373-007-0760-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-007-0760-x