IJPAM: Volume 73, No. 3 (2011)

GROUP DIVISIBLE DESIGNS WITH
TWO ASSOCIATE CLASSES AND $\lambda_2=4$

Chariya Uiyyasathian$^1$, Nittiya Pabhapote$^2$
$^1$Department of Mathematics
Faculty of Science
Chulalongkorn University
Bangkok, 10330, THAILAND
$^1$Center of Excellence in Mathematics, CHE
Sri Ayutthaya Rd., Bangkok, 10400, THAILAND
$^2$School of Science and Technology
University of the Thai Chamber of Commerce
Dindaeng, Bangkok, 10400, THAILAND


Abstract. A group divisible design GDD $(v=v_1+v_2+\dots +v_g, g, k, \lambda_1, \lambda_2)$ is an ordered triple $(V, G, \B),$ where $V$ is a $v$-set of symbols, $G$ is a partition of $V$ into $g$ sets of size $v_1, v_2, \dots, v_g$, each set being called group, and $\B$ is a collection of $k$-subsets (called blocks) of $V$, such that each pair of symbols from the same group occurs in exactly $\lambda_1$ blocks; and each pair of symbols from different groups occurs in exactly $\lambda_2$ blocks. Here, we focus on an existence problem of GDDs with two associate classes or when $g=2$, and with blocks of size 3, when the required designs have two groups of unequal sizes and $\lambda_2=4$. We obtain the necessary conditions and prove that these conditions are sufficient.

Received: August 19, 2011

AMS Subject Classification: 05B05, 05B07

Key Words and Phrases: graph decomposition, group divisible design

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 73
Issue: 3