The 3-GDDs of type $g^3u^2$

Yıl 2016, Cilt: 3 Sayı: 3, 135 - 144, 09.08.2016

Charles J. Colbourn Melissa S. Keranen Donald L. Kreher

https://doi.org/10.13069/jacodesmath.52790

Öz

A 3-GDD of type ${g^3u^2}$ exists if and only if  $g$ and $u$ have the same parity, $3$ divides $u$ and $u\leq 3g$.Such a 3-GDD of type ${g^3u^2}$ is  equivalent to an edge  decomposition of $K_{g,g,g,u,u}$ into triangles.

Kaynakça

• D. Bryant, D. Horsley, Steiner triple systems with two disjoint subsystems, J. Combin. Des. 14(1) (2006) 14–24.
• C. J. Colbourn, Small group divisible designs with block size three, J. Combin. Math. Combin. Comput. 14 (1993) 153–171.
• C. J. Colbourn, C. A. Cusack, D. L. Kreher, Partial Steiner triple systems with equal-sized holes, J. Combin. Theory Ser. A 70(1) (1995) 56–65.
• C. J. Colbourn, J. H. Dinitz (Eds.), Handbook of Combinatorial Designs, Second Edition, CRC/Chapman and Hall, Boca Raton, FL, 2007.
• C. J. Colbourn, D. Hoffman, R. Rees, A new class of group divisible designs with block size three, J. Combin. Theory Ser. A 59(1) (1992) 73–89.
• C. J. Colbourn, M. A. Oravas, R. S. Rees, Steiner triple systems with disjoint or intersecting subsystems, J. Combin. Des. 8(1) (2000) 58–77.
• R. Rees, Uniformly resolvable pairwise balanced designs with blocksizes two and three, J. Combin. Theory Ser. A 45(2) (1987) 207-225.
• R. M. Wilson, An existence theory for pairwise balanced designs. I. Composition theorems and morphisms, J. Combinatorial Theory Ser. A 13 (1972) 220–245.
• R. M. Wilson, An existence theory for pairwise balanced designs. II. The structure of PBD-closed sets and the existence conjectures, J. Combinatorial Theory Ser. A 13 (1972) 246–273.