Abstract
A group divisible design GDD(v = 1 + n + n,3, 3, λ 1, λ 2) is an ordered pair \((V, \mathcal{B})\) where V is an (1 + n + n)-set of symbols and \(\mathcal{B}\) is a collection of 3-subsets (called blocks) of V satisfying the following properties: the (1 + n + n)-set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ 1 blocks in \(\mathcal{B}\); and each pair of symbols from different groups occurs in exactly λ 2 blocks in \(\mathcal{B}\). Let λ 1, λ 2 be positive integers. Then the spectrum of λ 1, λ 2, denoted by Spec(λ 1, λ 2), is defined by
Spec(λ 1, λ 2) = {n ∈ ℕ: a GDD(v = 1 + n + n,3, 3, λ 1, λ 2) exists}.
We found in [10] the spectrum Spec(λ 1, λ 2) provided that λ 1 ≥ λ 2 in all situations. We find in this paper Spec(λ 1, λ 2) when λ 1 < λ 2 in all situations.
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Lapchinda, W., Punnim, N., Pabhapote, N. (2013). GDDs with Two Associate Classes and with Three Groups of Sizes 1, n, n and λ 1 < λ 2 . In: Akiyama, J., Kano, M., Sakai, T. (eds) Computational Geometry and Graphs. TJJCCGG 2012. Lecture Notes in Computer Science, vol 8296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45281-9_10
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DOI: https://doi.org/10.1007/978-3-642-45281-9_10
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