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Kirkman packing and covering designs with spanned holes of size 2. (English) Zbl 1062.05035
Summary: A Kirkman holey packing (resp. covering) design, denoted by KHPD\((g^u)\) (resp. KHCD\((g^u)\)), is a resolvable (\(gu\), 3, 1) packing (resp. covering) design of pairs with \(u\) disjoint holes of size \(g\), which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size \(\delta\), while other blocks have size 3. Here \(\delta\) is equal to 2, 3, and 4 when \(gu \equiv 2, 3\), and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD\((2^u)\) and a KHCD\((2^u)\) is solved with one possible exception of a KHPD(\(2^8\)).

MSC:
05B40 Combinatorial aspects of packing and covering
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