Uniformly resolvable pairwise balanced designs with blocksizes two and three

Decompositions of complete graphs into triangles and Hamilton cycles

“…Recently, Horak et al [15] solved the problem of finding 2-factorizations of K n into triangle factors and hamilton cycles for several infinite classes of orders n (a triangle factor is a 2-factor in which each component is a triangle). In 1987, Rees [16] proved necessary and sufficient conditions for a decomposition of K n into triangle factors and 1-factors. There is a paper by Dejter et al [11] which is concerned with the total number of triangles in 2-factorizations of K n .…”

Group Divisible Designs with Two Associate Classes:n=2 orm=2

“…Theorem 2.7 [15]. For all n#0 (mod 6) and for all even x with 0 x<n except for (n, x) # [ (12,10), (6,4)], there exists an x-regular simple graph H on n vertices whose edges can be resolvably partitioned into triples, such that K n &E(H ) has a 1-factorization.…”

“…In particular, one technique developed here requires knowing when there exists a multigraph on n vertices whose edges can be partitioned into triples, and whose complement in *K n has a 1-factorization (see Theorem 2.9). This result is of interest in its own right (see [15], for example).…”

Two new direct product‐type constructions for resolvable group‐divisible designs

“…Theorem 3.7 was first proven by the author in a lengthy paper (see [8]); through Lemma 3.3 we are able to give a short and simple proof of this result by applying "weight 2" to Kirkman and Nearly Kirkman Triple Systems. Now for each i = r + 1, .…”