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The existence of Kirkman squares – doubly resolvable \((v,3,1)\)-BIBDs. (English) Zbl 0995.05021
A Kirkman square with index \(\lambda\), latinicity \(\mu\), block size \(k\), and \(v\) points, KS\(_k(v;\mu ,\lambda)\), is a \(t\times t\) array (\(t = \lambda (v-1)/\mu (k-1)\)) defined on a \(v\)-set \(V\) such that (1) every point of \(V\) is contained in precisely \(\mu\) cells of each row and column, (2) each nonempty cell of the array contains a \(k\)-subset of \(V\), and (3) the collection of blocks from all cells form a \((v,k,\lambda)\)-BIBD. The existence of a KS\(_k(v;1,\lambda)\) is equivalent to the existence of a doubly resolvable \((v,k,\lambda)\)-BIBD.
The spectrum of KS\(_2(v;1,1)\), also known as Room square, was completed by R. C. Mullin and W. D. Wallis [Aequationes Math. 13, 1-7 (1975; Zbl 0315.05014)]. For \(k\geq 3\), the spectrum has been determined for KS\(_3(v;1,2)\) [J. Comb. Theory, Ser. A 72, 50-76 (1995; Zbl 0836.05008)] and KS\(_3(v;2,4)\) [Discrete Math. 186, 195-216 (1998; Zbl 0956.05027)], both by E. R. Lamken. Here the spectrum of KS\(_3(v;1,1)\) is determined; these exist for \(v = 6n+3\), \(n\) a non-negative integer, except for \(n \in \{1,2\}\), and with the following values of \(n\) still open: 3, 9, 11, 15, 16, 17, 19, 23, 24, 25, 27, 29, 30, 31, 33, 38, 39, 41, 43, 44, 47, 58, 59.

MSC:
05B07 Triple systems
05B05 Combinatorial aspects of block designs
05B15 Orthogonal arrays, Latin squares, Room squares
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