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Resolvable Mendelsohn designs with block size 4. (English) Zbl 0725.05012

A set of v points together with a collection of k-cycles (called blocks) described on the points is a (v,k,\(\lambda\))-Mendelsohn design if any two points are consecutive in exactly \(\lambda\) blocks. If any two points are at a distance t in exactly \(\lambda\) blocks for \(t=1,2,...,k-1\) then the design is said to be perfect. A (v,k,\(\lambda\))-Mendelsohn design is resolvable if either \(v\equiv 0(mod k)\) and the set of blocks can be partitioned into classes of v/k blocks which cover all the points, or \(v\equiv 1(mod k)\) and the blocks can be partitioned into classes of (v- 1)/k blocks which cover all but one point (for the latter case the term “almost resolvable” has also been used). Here the authors construct a resolvable (v,4,1)-Mendelsohn design for all \(v\equiv 0(mod 4)\) except \(v=4\) and \(v=12\). For \(v\equiv 1(mod 4)\) and \(v\neq 57\), 93 they construct a resolvable design which is also perfect, slightly improving a known result. They construct a resolvable design for \(v=57\), 93 and in a note added in proof they announce the existence of perfect examples also in these two case.

MSC:

05B05 Combinatorial aspects of block designs
05C20 Directed graphs (digraphs), tournaments
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References:

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