New Z-cyclic triplewhist frames and triplewhist tournament designs

Triplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on vv players is nearly complete. It is now known that triplewhist designs do not exist for v=5,9,12,13v=5,9,12,13 and do exist for all other v≡0,1(mod4) except, possibly, v=17v=17. Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z  -cyclic if the players are elements in Zm∪AZm∪A where m=vm=v, A=∅A=∅ when v≡1(mod4) and m=v-1m=v-1, A={∞}A={∞} when v≡0(mod4) and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R1,R2,…R1,R2,… in such a way that Rj+1Rj+1 is obtained by adding +1(modm) to every element in RjRj. The production of Z-cyclic triplewhist designs is particularly challenging when m   is divisible by any of 5,9,11,13,175,9,11,13,17. Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m   being divisible by at least one of 5,9,11,13,175,9,11,13,17.