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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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