Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

Abstract

Abstract:The necessary conditions for the existence of a super-simple resolvable balanced incomplete block design on v v points with k ¼ 4 and ¼ 3, are that v v ! 8 and v v 0 mod 4. These conditions are shown to be sufficient except for v v ¼ 12 . #

Ordered tournaments and ordered triplewhist tournaments with the three person property

“…49 (42, 40, 39, 24), (27,48,44,3), (23,45,10,16), (46,37,28,12), (8,26,31,18), (19,15,38,13), (43,33,21,32), (36,41,25,6), (20,34,5,22), (2, 1, 9, 29), (4,11,14,17), (7,30,47,35). 57 (25,14,1,56), (10,31,26,40), (2, 35, 52, 39), (9,47,19,11), (29,38,4,…”

“…The initial round of games is obtained by multiplying the block (x, x 2 , x 31 = 2, x 84 = 4x + 4) by x 4t for 0 ≤ t ≤ 30. (18,10,19,26), (7,16,31,12), (21,20,27,9), (15,25,28,11), (1,13,29,23), (24,22,3,14), (30,17,4,32). 45 (1,3,5,21), (4,43,6,34), (33,41,14,29), (7,25,32,10), (44,12,31,28), (13,17,8,18), (22,<...>…”

“…45 (1,3,5,21), (4,43,6,34), (33,41,14,29), (7,25,32,10), (44,12,31,28), (13,17,8,18), (22,2,15,39), (40,9,19,26), (38,37,16,27), (35,30,24,36), (20,11,23,42). 49 (42, 40, 39, 24), (27,48,44,3), (23,45,10,16), (46,37,28,…”

Existence of directedwhist tournaments with the three person property 3PDWh(v)

“…This kind of design was introduced and studied by Gronau and Mullin [17] and also studied by Chen [9,10]. Such designs with resolvable property were investigated by Ge and Lam [15]. For the existence of 3PWh(v)s, Finizio [11][12][13] obtained several infinite classes and some examples.…”

Existence of Z-cyclic 3PDTWh(p) for Prime p ≡ 1 (mod 4)

“…This kind of design was introduced and studied by Gronau and Mullin [29] and also studied by Chen [15,16]. Such designs with resolvable property were investigated by Ge and Lam [25] and Zhang and Ge [38].…”