J Austral Math Soc Ser A 44 pp33--41, 1988.

The Existence of a Class of Kirkman Squares of Index 2

E. R. Lamken and S. A. Vanstone

Abstract

A Kirkman square with index l, latinicity m, block size k and v points, KS_k(v; m, l), is a t ×t array (t = l(v - 1)/m(k - 1)) defined on a v-set V such that (1) each point of V is contained in precisely m cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (v, k, l)-BIBD. For m = 1, the existence of a KS_k(v; m, l) is equivalent to the existence of a doubly resolvable (v, k, l)-BIBD. In this case the only complete results are for k = 2. The case k = 3, l = 1 appears to be quite difficult although some existence results are available. For k = 3, l = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS₃(v; 1, 2) for all v º 3 (mod 12).

1980 AMS Subject Classification: 05B30

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Authors

E. R. Lamken

S. A. Vanstone

Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N21 3G1, Canada.