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

On the existence of nearly Kirkman triple systems with subsystems

“…5. For v = 228, take a TD (8,11) and give all points on the first six groups weight 6 and all points on a seventh group weight 18; then give two points on the last group weight 12 and the remaining nine points on that group weight 6 to obtain a Kirkman frame of type 66 6 198 1 78 1 . Adjoin 30 ideal points and apply Construction 2.…”

“…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 …”

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] .…”

Embeddings of resolvable group divisible designs with block size 3

“…An RGD (3,1; v) is called a Kirkman triple system and denoted by KTS(v). An RGD (3,2; v) is called a nearly Kirkman triple system and denoted by NKTS(v).…”

“…An RGD (3,2; v) is called a nearly Kirkman triple system and denoted by NKTS(v). Theorem 1.1 [1,[12][13][14] An RGD (3, g; v) exists if and only if v ≥ 3g, v − g ≡ 0 (mod 2), v ≡ 0 (mod 3), v ≡ 0 (mod g), and (g, v) ∈ {(2, 6), (2,12), (6,18)}. Now let (X 1 , G 1 , B 1 ) be an RGD(K, M; v), and (X 2 , G 2 , B 2 ) be an RGD(K, M; u).…”