Resolvable packings of K v with K 2 × K c 's

The study of resolvable packings of K v with K r  ×  K c 's is motivated by the use of DNA library screening. We call such a packing a ( v , K r  ×  K c , 1)‐RP. As usual, a ( v , K r  ×  K c , 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v  ≡ 0 (mod rc ). Let ρ be the number of parallel classes of a ( v , K r  ×  K c , 1)‐RP. Then we have ρ ≤ ⌊( v ‐1)/( r  +  c  − 2)⌋. In this article, we present a number of constructive methods to obtain optimal ( v, K 2  ×  K c , 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal ( v, K 2  ×  K 3 , 1)‐RP meeting the bound exists if and only if v  ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009