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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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