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 v ≡ 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 v = 12 and except possibly when v = 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, v ≡ 0 ( mod 12 ) , v ≠ 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, v ≡ 0 ( mod 12 ) , v ≠ 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.