IJPAM: Volume 104, No. 1 (2015)

SOME CONSTRUCTION OF
GROUP DIVISIBLE DESIGNS $\GDD(m,n;1,3)$

Chariya Uiyyasathian$^1$, Narong Punnim$^2$
$^1$Department of Mathematics and Computer Science
Faculty of Science
Chulalongkorn University
Payathai Rd., Bangkok 10330, THAILAND
$^2$Department of Mathematics
Srinakharinwirot University
Sukhumvit 23, Bangkok 10110, THAILAND


Abstract. A group divisible design $\GDD(m,n;1,3)$ is an ordered pair $(V, \B)$ where $V$ is an $(m+n)$-set of symbols and $\B$ is a collection of $3$-subsets (called blocks) of $V$ satisfying the following properties: the $(m+n)$-set is divided into two groups of size $m$ and $n$; each pair of symbols from the same group occurs in exactly one block in $\B$; and each pair of symbols from different groups occurs in exactly three blocks in $\B$. Given positive integers $m$ and $n$, two necessary conditions on $m$ and $n$ for the existence of a $\GDD(m,n;1,3)$ are $ 6\mid [m(m-1)+ n(n-1)]$ and $m\not\equiv n (\mod 2)$. We show that these conditions are sufficient for the most cases.

Received: March 23, 2015

AMS Subject Classification:

Key Words and Phrases: group divisible design, difference triple, graph decomposition

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DOI: 10.12732/ijpam.v104i1.2 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 104
Issue: 1
Pages: 19 - 28


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