Note on three table Oberwolfach problem

The well known Oberwolfach problem asks the existence of a 2-factorization of the complete graph KnKn (when n is odd) or Kn−IKn−I (when n is even) in which each 2-factor is isomorphic to the given 2-factor F  . The notation OP(m1α1,m2α2,…,mtαt) represents the Oberwolfach problem in which each 2-factor consists of exactly αiαi cycles of length mimi, i=1,2,…,ti=1,2,…,t. In this paper, we show that the following exists: (i) OP(5,6,4s)OP(5,6,4s), OP(5,10,4s)OP(5,10,4s), OP(5,14,4s)OP(5,14,4s), OP(5,6,4s,4s)OP(5,6,4s,4s), OP(5,14,4s−4,4s−4)OP(5,14,4s−4,4s−4), OP(5,4s,4s+6)OP(5,4s,4s+6), OP(5,4s−4,4s+10)OP(5,4s−4,4s+10) and OP(7,4,4s)OP(7,4,4s) for all s≥3s≥3. (ii) OP2(s+1,st,st)OP2(s+1,st,st) for all t≥2t≥2 and odd s≥3s≥3. As a consequence, number of unknown cases of the Oberwolfach problem getting reduced.