Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T14:19:53.186Z Has data issue: false hasContentIssue false
  • English
  • Français

An Existence Theory for Incomplete Designs

Published online by Cambridge University Press:  20 November 2018

Peter Dukes ,
Esther R. Lamken  and
Alan C. H. Ling
Peter Dukes
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC e-mail: [email protected]
Esther R. Lamken
Affiliation:
66h Colby Street, San Francisco, CA, USA 94134 e-mail: [email protected]
Alan C. H. Ling
Affiliation:
Department of Computer Science, University of Vermont, Burlington, VT, USA 05405 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size $w$ , and all (other) block sizes equal $k$ , this is denoted $\text{IPBD}\left( \left( v;w \right),\,k \right)$ . In addition to congruence restrictions on $v$ and $w$ , there is also a necessary inequality: $v\,>\,\left( k\,-\,1 \right)w$ . This article establishes two main existence results for $\text{IPBD}\left( \left( v;w \right),\,k \right)$ : one in which $w$ is fixed and $v$ is large, and the other in the case $v>\,\left( k-1+\varepsilon \right)w$ when $w$ is large (depending on $\varepsilon$ ). Several possible generalizations of the problemare also discussed.

Keywords

05C70block designhypergraph
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 287 - 302
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Chan, J. H., Dukes, P. J., Lamken, E. R., and Ling, A. C. H., The asymptotic existence of resolvable group divisible designs. J. Combin. Des. 21(2013), 112–126.Google Scholar
[2] Chowla, S., P. Erdôs, and Strauss, E. G., On the maximal number ofpairwise orthogonal latinsquares of a given order. Canad. J. Math. 12(1960), 204–208. http://dx.doi.org/10.4153/CJM-1960-017-2 Google Scholar
[3] Colbourn, C. J. and Rosa, A., Triple systems.Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999.Google Scholar
[4] Doyen, J. and Wilson, R. M., EmbeddingsofSteiner triple systems.Discrete Math. 5(1973), 229–239. http://dx.doi.org/10.1016/0012-365X(73)90139-8 Google Scholar
[5] Draganova, A., Asymptotic existence of decompositions of edge-colored graphs and hypergraphs. Ph.D. dissertation, University of California, Los Angeles, 2006.Google Scholar
[6] Dukes, P.J. and Ling, A. C. H., Asymptotic existence of resolvable graph designs. Canad.Math. Bull. 50(2007), no. 4, 504–518. http://dx.doi.org/10.41 53/CMB-2OO7-O5O-X Google Scholar
[7] Gustavsson, T., Decompositions of large graphs and digraphs with high minimum degree.Doctoral Dissertation, Department of Mathematics, Stockholm University, 1991.Google Scholar
[8] Keevash, P., The existence of designs. arxiv:1401.3665v1.Google Scholar
[9] Lamken, E. R. and Wilson, R. M., Decompositions of edge-colored complete graphs. J. Combin. Theory Ser.A 89(2000), no. 2,149-200. http://dx.doi.org/10.1006/jcta.1999.3005 Google Scholar
[10] Liu, J., Asymptotic existence theorems for frames and group divisible designs.J. Combin. Theory Ser. A 114(2007), no. 3, 410–420. http://dx.doi.0rg/IO.IOI6/j.jcta.2OO6.O6.OO6.Google Scholar
[11] Mendelsohn, E. and Rosa, A., Embedding maximal packings of triples.Congr.Numer. 40(1983), 235–247.Google Scholar
[12] Ray-Chaudhuri, D. K. Wilson, and R.M., The existence of resolvable block designs. In: A Survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), North-Holland, Amsterdam, 1973, pp. 361–376.Google Scholar
[13] Rees, R. and Stinson, D. R., On combinatorial designs with subdesigns. Combinatorial designsâÂîatribute to HaimHanani.Discrete Math. 77(1989), no. 1-3, 259–279. http://dx.doi.org/1 0.101 6/0012-365X(89)90365-8 Google Scholar
[14] Rees, R. and Stinson, D. R., On the existence of incomplete designs of block size four having one hole. Utilitas Math. 35(1989), 119–152.Google Scholar
[15] Stinson, D. R., A new proof of the Doyen-Wilson theorem. J. Austral. Math. Soc. Ser. A 47(1989), no. 1, 32–42. http://dx.doi.org/10.1017/S1446788700031177 Google Scholar
[16] van Bommel, C. M., An asymptotic existence theory on incomplete mutually orthogonal latin squares. M.Sc. thesis, University of Victoria, 2015.Google Scholar
[17] Wilson, R. M., An existence theory for pairwise balanced designs. II. The structure of PBD-closed sets and the existence conjectures. J. Combin. Theory Ser. A 13(1972), 246–273. http://dx.doi.org/10.1016/0097-3165(72)90029-5 Google Scholar
[18] Wilson, R. M., An existence theory for pairwise balanced designs. III. Proof of the existence conjectures. J. Combin.Theory Ser.A 18(1975), 71–79. http://dx.doi.org/10.1016/0097-3165(75)90067-9.Google Scholar
[19] Wilson, R. M., The construction of group divisible designs and partial planes having the maximum number of lines of a given size. In: Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications, Univ. North Carolina, Chapel Hill, N.C. 1970, pp. 488–497.Google Scholar
[20] Wilson, R. M., Constructions and uses ofpairwise balanced designs. In: Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974) Part 1: Theory of designs, finite geometry and coding theory, Math. Centre Tracts, 55, Math. Centrum, Amsterdam, 1974, pp. 18–41.Google Scholar
[21] Wilson, R. M., Decompositions of complete graphs into subgraphs isomorphic to a given graph. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), CongressusNumerantium, 15, Utilitas Math., Winnipeg, Man., 1976, pp. 647–659. Department of Mathematics and Statistics, University of Victoria, Victoria, BC e-mail: [email protected]Google Scholar