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

It is proved in this paper that for h is an element of {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 equivalent to 0 (mod 6) and v greater than or equal to 3h, thus providing a good foundation for the embedding problem for NKTS. As an application, we prove that: (1) For u equivalent to v equivalent to 0 (mod 6), v greater than or equal to 48, and u greater than or equal to 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 equivalent to 0 (mod 6) and u greater than or equal to 3v. (C) 2002 Elsevier Science B.V. All rights reserved.