On resolvable designs

A survey on the existence ofG‐Designs

“…Clearly, if t|V (G)| divides n, then the existence of a resolvable G-design of order n implies the existence of an H-design of order n where H is the union of t vertex-disjoint copies of G. Numerous results on the existence of G-designs are thus obtained from results on the existence of resolvable G-designs. Such results have been obtained for complete graphs [73,103] (also see [4]), cycles [23], trees [96,120], paths [27], stars [120], cubes [18], complete bipartite graphs [81], and for K 4 − e (the complete graph on four vertices with an edge removed) [51]. Also see the survey [39].…”

Block designs with large holes and α ‐resolvable BIBDs

“…As a ranges over GF(u), the required frame is obtained. Proof: As noted, it is shown in [5] Let u be an integer such that u = 1 or 5 (mod 10) and u 2 21. Then there Proof.…”

On the existence of triplewhist tournaments TWh(v)

“…For each n, all the holey rounds will be generated modulo 4n from the following initial holey round with hole S 0 : n = 6, (1, 2, 3, 10), (4,17,19,14), (5,9,22,8), (7,15,20,11), (16,13,21,23); Proof. Let S = GF (49) and x be a primitive root satisfying f (x) = x 2 + x + 3 = 0.…”