Group divisible designs of four groups and block size five with configuration (1, 1, 1, 2)

Abstract

We present constructions and results about GDDs with four groups and block size five in which each block has Configuration $(1, 1, 1, 2)$, that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD$(n, 4, 5; \lambda_1, \lambda_2)$ with Configuration $(1, 1, 1, 2)$, and show that the necessary conditions are sufficient for a GDD$(n, 4, 5; \lambda_1,$ $\lambda_2)$ with Configuration $(1, 1, 1, 2)$ if $n \not \equiv 0 ($mod $6)$, respectively. We also show that a GDD$(n, 4, 5; 2n, 6(n - 1))$ with Configuration $(1, 1, 1, 2)$ exists, and provide constructions for a GDD$(n = 2t, 4, 5; n, 3(n - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 12$, and a GDD$(n = 6t, 4, 5; 4t, 2(6t - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 6$ and $18$, respectively.

Keywords

• Group divisible designs (GDDs)
• Latin squares
• Block configurations
• 1-factors
• RGDDs
• RBIBDs

Kaynakça

1. C. J. Colbourn, D. H. Dinitz (Eds.), Handbook of Combinatorial Designs, Second Edition, Chapman and Hall, CRC Press, Boca Raton, FL, 2007.
2. C. J. Colbourn, A. Rosa, Triple System, Oxford Science Publications, Clarendon Press, Oxford, 1999.
3. H. L. Fu, C. A. Rodger, Group divisible designs with two associate classes: n = 2 or m = 2, J. Combin. Theory Ser. A 83(1) (1998) 94–117.
4. H. L. Fu, C. A. Rodger, D. G. Sarvate, The existence of group divisible designs with first and second associates, having block size three, Ars Combin. 54 (2000) 33–50.
5. G. Ge, A. C. H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13(3) (2005) 222–237.
6. D. Henson, D. G. Sarvate, S. P. Hurd, Group divisible designs with three groups and block size four, Discrete Math. 307(14) (2007) 1693–1706.
7. S. P. Hurd, N. Mishra, D. G. Sarvate, Group divisible designs with two groups and block size five with fixed block configuration, J. Combin. Math. Comput. 70 (2009) 15–31.
8. S. P. Hurd, D. G. Sarvate, Odd and even group divisible designs with two groups and block size four, Discrete Math. 284(1-3) (2004) 189–196.

9. S. P. Hurd, D. G. Sarvate, Group divisible designs with block size four and two groups, Discrete Math. 308(13) (2008) 2663–2673.
10. M. S. Keranen, M. R. Laffin, Fixed block configuration group divisible designs with block size six, Discrete Math. 308(4) (2012) 745–756.
11. C. C. Lindner, C. A. Rodger, Design Theory, Second edition, CRC Press, Boca Raton, 2008.
12. E. Lucas, Récréations Mathématiques, Vol. 2, Gauthier-Villars, Paris, 1883.
13. R. C. Mullin, H. O. F. Gronau, PBDs and GDDs: the basics, C. J. Colbourn J. H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL (1996), 185–193.
14. N. Punnim, D. G. Sarvate, A construction for group divisible designs with two groups, Congr. Numer. 185 (2007) 57–60.
15. R. S. Rees, Two new direct product-type constructions for resolvable group-divisible designs, J. Combin. Des. 1(1) (1993) 15–26.
16. D. G. Sarvate, S. P. Hurd, Group divisible designs with two groups and block configuration (1; 4), J. Combin. Inform. System Sci. 32 (2007) 297–306.
17. A. P. Street, D. J. Street, Combinatorics of Experimental Design, Clarendon Press, Oxford, 1987.
18. A. P. Street, D. J. Street, Partially balanced incomplete block designs, C. J. Colbourn J. H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL (1996), 419–423.
19. M. Zhu, G. Ge, Mixed group divisible designs with three groups and block size four, Discrete Math. 310(17-18) (2010) 2323–2326.