Uniformly resolvable designs with index one and block sizes three and four - with three or five parallel classes of block size four

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size. A URD with v points and with block sizes three and four means that at least one parallel class has block size three and at least one has block size four. Danziger [P. Danziger, Uniform restricted resolvable designs with r = 3, ARS Combin. 46 (1997) 161-176] proved that for all upsilon 12 (mod 24) there exist URDs with index one, some parallel classes of block size three, and exactly three parallel classes with block size four, except when upsilon = 12 and except possibly when upsilon = 84 156. We extend Danziger's work by showing that there exists a URD with index one, some parallel classes with block size three, and exactly three parallel classes with block size four if, and only if, upsilon 0 (mod 12), upsilon not equal 12. We also prove that there exists a URD with index one, some parallel classes of block size three, and exactly five parallel classes with block size four if, and only if, upsilon 0 (mod 12), upsilon not equal 12. New labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs. Some ingredient URDs are also constructed with difference families. (c) 2008 Elsevier B.V. All rights reserved.