Group Divisible Designs with Block Size Four and Type $$g^u b^1 (gu/2)^1$$ g u b 1 ( g u / 2 ) 1
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ORIGINAL PAPER
Group Divisible Designs with Block Size Four and Type gu b1 (gu=2Þ1 Anthony D. Forbes1 Received: 19 September 2019 / Revised: 27 March 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract We discuss group divisible designs with block size four and type gu b1 ðgu=2Þ1 , where u ¼ 5, 6 and 7. For integers a and b, we prove the following. (i) A 4-GDD of type ð4aÞ5 b1 ð10aÞ1 exists if and only if a 1, b a (mod 3) and 4a b 10a. (ii) A 4-GDD of type ð6a þ 3Þ6 b1 ð18a þ 9Þ1 exists if and only if a 0, b 3 (mod 6) and 6a þ 3 b 18a þ 9. (iii) A 4-GDD of type ð6aÞ6 b1 ð18aÞ1 exists if and only if a 1, b 0 (mod 3) and 6a b 18a. (iv) A 4-GDD of type ð12aÞ7 b1 ð42aÞ1 exists if and only if a 1, b 0 (mod 3) and 12a b 42a, except possibly for 12a 2 f120; 180; 240; 360; 420; 720; 840g, 24a\b\42a, for 12a 2 f144; 1008g, 30a\b\42a, and for 12a 2 f168; 252; 336; 504; 1512g, 36a\b\42a. Keywords Group divisible design 4-GDD Wilson’s fundamental construction
Mathematics Subject Classification 05B05
1 Introduction A group divisible design, K-GDD, of type gu11 . . .gur r is an ordered triple (V; G; B) such that: (i) (ii) (iii)
V is a base set of cardinality u1 g1 þ þ ur gr ; G is a partition of V into ui subsets of cardinality gi , i ¼ 1; . . .; r, called groups; B is a non-empty collection of subsets of V with cardinalities k 2 K, called blocks; and
& Anthony D. Forbes [email protected] 1
School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
123
Graphs and Combinatorics
(iv)
each pair of elements from distinct groups occurs in precisely one block but no pair of elements from the same group occurs in any block.
We often abbreviate fkg-GDD to k-GDD, and a k-GDD of type qk is also called a transversal design, TD(k, q). A parallel class in a group divisible design is a subset of the block set in which each element of the base set appears exactly once. A kGDD is called resolvable, and is denoted by k-RGDD, if the entire set of blocks can be partitioned into parallel classes. If there exist k mutually orthogonal Latin squares (MOLS) of side q, then there exists a ðk þ 2Þ-GDD of type qkþ2 and a ðk þ 1ÞRGDD of type qkþ1 [2, Theorem III.3.18]. Furthermore, as is well known, there exist q 1 MOLS of side q whenever q is a prime power. Group divisible designs are widely used Design Theory, especially in the construction of infinite classes of combinatorial designs by means of a technique known as Wilson’s Fundamental Construction [31, 17, Theorem IV.2.5]. In general, the existence spectrum problem for group divisible designs with constant block sizes, k-GDDs, appears to be a long way from being completely solved, except of course when k ¼ 2 in which case the design is essentially a complete multipartite graph. Nevertheless, for k ¼ 3 and 4, provided there are not too many distinct group sizes, considerable progress has been made. Indeed, the spectrum has been completely determ
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