On the existence and application of incomplete nearly Kirkman triple systems with a hole of size 6 or 12 Original Research Article

It is proved in this paper that for h∈{6,12}, there exists an incomplete nearly Kirkman triple system (NKTS) of order v having a hole of size h if and only if v≡0 (mod 6) and v⩾3h, thus providing a good foundation for the embedding problem for NKTS. As an application, we prove that: (1) For u≡v≡0 (mod 6), v⩾48, and u⩾4v−18, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36 or 42, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0 (mod 6) and u⩾3v.