Some new triplewhist tournaments TWh(v)

It was Moore who first introduced the triplewhist tournament TWh(v) problem in 1896. It is proved in the literature that the necessary condition for the existence of a TWh(v), namely, v≡0 or 1 (mod 4), is also sufficient except for v=5,9 and possibly excepting v∈{12,56}∪{13,17,45,57,65,69,77,85,93,117,129,153}. In this paper, it is shown that there is no TWh(12) and that there does exist a Z-cyclic TWh(v) for each v∈{44,45,48,52,56}. This completes the even case for the existence of TWh(v). By applying frame constructions and product constructions, several new infinite classes of Z-cyclic triplewhist tournaments are then obtained.