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

Yıl 2016, , 187 - 194, 09.08.2016

Ronald Mwesigwa Dinesh G. Sarvate Li Zhang

https://doi.org/10.13069/jacodesmath.70490

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• C. J. Colbourn, D. H. Dinitz (Eds.), Handbook of Combinatorial Designs, Second Edition, Chapman and Hall, CRC Press, Boca Raton, FL, 2007.
• C. J. Colbourn, A. Rosa, Triple System, Oxford Science Publications, Clarendon Press, Oxford, 1999.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• S. P. Hurd, D. G. Sarvate, Group divisible designs with block size four and two groups, Discrete Math. 308(13) (2008) 2663–2673.
• M. S. Keranen, M. R. Laffin, Fixed block configuration group divisible designs with block size six, Discrete Math. 308(4) (2012) 745–756.
• C. C. Lindner, C. A. Rodger, Design Theory, Second edition, CRC Press, Boca Raton, 2008.
• E. Lucas, Récréations Mathématiques, Vol. 2, Gauthier-Villars, Paris, 1883.
• 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.
• N. Punnim, D. G. Sarvate, A construction for group divisible designs with two groups, Congr. Numer. 185 (2007) 57–60.
• R. S. Rees, Two new direct product-type constructions for resolvable group-divisible designs, J. Combin. Des. 1(1) (1993) 15–26.
• 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.
• A. P. Street, D. J. Street, Combinatorics of Experimental Design, Clarendon Press, Oxford, 1987.
• 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.
• M. Zhu, G. Ge, Mixed group divisible designs with three groups and block size four, Discrete Math. 310(17-18) (2010) 2323–2326.