Abstract
A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . In this paper we consider the case of block sizes 3 and 4. The cases r 3 = 1 and r 4 = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ≡ 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r 3 = 4 exist, and that those with r 3 = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r 3 ≤ 10. Furthermore, we prove that URDs with r 4 = 7 exist and that those with r 4 = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 142, 2 346 and 6 54. Finally, we provide four {3,5}-URDs with 105 points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abel R.J.R., Ge G., Greig M., Ling A.C.H. (2009) Further results on (V, {5, w *}, 1)-PBDs. Discrete Math. 309: 2323–2339
Abel R.J.R., Colbourn C.J., Dinitz J.H. (2006) Mutually orthogonal Latin squares (MOLS). In: Colbourn C.J., Dinitz J.H. (eds) The CRC handbook of combinatorial designs, 2 edn. CRC Press, Boca Raton, FL, pp 160–193
Beth T., Jungnickel D., Lenz H. (1999) Design theory, 2nd edn. Cambridge University Press, Cambridge
Colbourn, C.J., Dinitz, J.H. (eds) (2006) The CRC handbook of combinatorial designs, 2nd edn. CRC Press, Boca Raton FL
Danziger P. (1997) Uniform restricted resolvable designs with r = 3. Ars. Combin. 46: 161–176
Danziger P., Mendelsohn E. (1996) Uniformly resolvable designs. JCMCC 21: 65–83
Dinitz J.H., Ling A.C.H., Danziger P. (2009) Maximum uniformly resolvable designs with block sizes 2 and 4. Discrete Math. 309: 4716–4721
Furino S.C., Miao Y., Yin J.X. (1996) Frames and resolvable designs: uses, constructions and existence. CRC Press, Boca Raton, Fl
Ge G. (2001) Uniform frames with block size four and index one or three. J. Combin. Des. 9: 28–39
Ge G. (2002) Resolvable group divisible designs with block size four. Discrete Math. 243: 109–119
Ge G. (2006) Resolvable group divisible designs with block size four and index three. Discrete Math. 306: 52–65
Ge G., Lam C.W.H. (2003) Resolvable group divisible designs with block size four and group size six. Discrete Math. 268: 139–151
Ge G., Ling A.C.H. (2004) A survey on resolvable group divisible designs with block size four. Discrete Math. 279: 225–245
Ge G., Ling A.C.H. (2005) Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs. J. Combin. Des. 13: 222–237
Ge G., Ling A.C.H. (2004) Some more 5-GDDs and optimal (v, 5, 1)-packings. J. Combin. Des. 12: 132–141
Ge G., Lam C.W.H., Ling A.C.H. (2004) Some new uniform frames with block size four and index one or three. J. Combin. Des. 12: 112–122
Ge G., Lam C.W.H., Ling A.C.H., Shen H. (2005) Resolvable maximum packings with quadruples. Des. Codes Cryptogr. 35: 287–302
Ge G., Rees R.S. (2002) On group-divisible designs with block size four and group-type. Des. Codes Cryptogr. 27: 5–24
Hanani H., Ray-Chauduri D.K., Wilson R.M. (1972) On resolvable designs. Discrete Math. 3: 343–357
Lamken E.R., Mills W.H., Rees R.S. (1998) Resolvable minimum coverings with quadruples. J. Combin. Des. 6: 431–450
Rees R.S. (1987) Uniformly resolvable pairwise balanced designs with blocksizes two and three. J. Combin. Theory Ser. A 45: 207–225
Rees R.S. (1993) Two new direct product-type constructions for resolvable group-divisible designs. J. Combin. Des. 1: 15–26
Rees R.S. (2000) Group-divisible designs with block size k having k+1 groups for k = 4, 5. J. Combin. Des. 8: 363–386
Rees R.S., Stinson D.R. (1992) Frames with block size four. Can. J. Math. 44: 1030–1049
Schuster E. (2009) Uniformly resolvable designs with index one and block sizes three and four—with three or five parallel classes of block size four. Discrete Math. 309: 2452–2465
Schuster E. (2009) Uniformly resolvable designs with index one, block sizes three and five and up to five parallel classes with blocks of size five. Discrete Math. 309: 4435–4442
Shen H. (1990) Constructions and uses of labeled resolvable designs. In: Wallis W.D. (eds) Combinatorial designs and applications. Marcel Dekker, New York, pp 97–107
Shen H. (1992) On the existence of nearly Kirkman systems. Ann. Discrete Math. 52: 511–518
Shen H., Wang M. (1998) Existence of labeled resolvable block designs. Bull. Belg. Math. Soc. 5: 427–439
Shen H., Shen J. (2002) Existence of resolvable group divisible designs with block size four I. Discrete Math. 254: 513–525
Sun X., Ge G. (2009) Resolvable group divisible designs with block size four and general index. Discrete Math. 309: 2982–2989
Zhang X., Ge G. (2007) On the existence of partitionable skew Room frames. Discrete Math. 307: 2786–2807
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Charles J Colbourn.
Electronic Supplementary Material
Rights and permissions
About this article
Cite this article
Schuster, E., Ge, G. On uniformly resolvable designs with block sizes 3 and 4. Des. Codes Cryptogr. 57, 45–69 (2010). https://doi.org/10.1007/s10623-009-9348-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-009-9348-1
Keywords
- Uniformly resolvable design
- Labeled uniformly resolvable design
- Resolvable group divisible design
- Frame
- Transversal design