Further results on nearly Kirkman triple systems with subsystems

On the existence of nearly Kirkman triple systems with subsystems

“…By adding 48 infinite points, we get the desired GDD. {4}-GDD of type 12 11 15 1 66 1 We construct a {3, 4}-GDD of type 12 11 {4}-GDD of type 12 12 15 1 72 1 We construct a {3, 4}-GDD of type 12 12 …”

“…5 [12] (1) For u ≡ v ≡ 0 (mod 6), v ≥ 78, and u ≥ 3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v = 18, 24, 30, 36, 42, 48, 54, 60, 66 or 72, there exists an NKTS(u) containing a sub-NKTS (v) if and only if u ≡ 0 (mod 6) and u ≥ 3v.…”

Embeddings of resolvable group divisible designs with block size 3

“…. , t 3 −t 4 2 , which partition X \C, and t 4 2 holey parallel classes, denoted by P C ,j , j = 1, 2, . .…”

“…Theorem 1.2 [15,16,19] A KTS(v) can be embedded in a KTS(u) if and only if u ≡ v ≡ 3 (mod 6), and u ≥ 3v. Theorem 1.3 [3,4,5,20] An NKTS(v) can be embedded in an NKTS(u) if and only if u ≡ v ≡ 0 (mod 6), u ≥ 3v, and v = 6, 12.…”

Construction and counting of 2t+1 Steiner triple

“…Wilson published a paper with topic Solution of Kirkman's schoolgirls problem to show how to construct Kirkman triple systems of order 3 6  n [3][4] . In 1961 a Chinese mathematician Lu Jiaxi posed the decomposable condition of BIBD design [5][6][7][8][9] .…”