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Resolvable minimum coverings with quadruples. (English) Zbl 0910.05019
A collection of $$k$$-element subsets (called blocks) out of a $$v$$-set such that each $$t$$-subset of the $$v$$-set is contained in at least one block is called a covering design, and it is said to be resolvable if the blocks can be partitioned into classes such that every element is contained in precisely one block of each class. For $$t=2$$, a resolvable minimum covering is denoted by RC$$(v,k)$$. A necessary condition for the existence of an RC$$(v,k)$$ is clearly $$v \equiv 0 \pmod k$$. For $$k=4$$, it is shown that this condition is sufficient except for $$v =12$$ and possibly for $$v \in \{104,108,116,132,156,164,204,212,228,276\}$$. The proof is obtained by several direct and recursive constructions of resolvable minimum coverings.

##### MSC:
 05B40 Combinatorial aspects of packing and covering 05B05 Combinatorial aspects of block designs
##### Keywords:
covering design; covering number; resolvable design
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