Embedding handcuffed designs with block size 2 or 3 in 4-cycle systems

Minimum embedding of a P4-design into a balanced incomplete block design of index λ

“…5. λ = 3, v = 10, w = 6. Blocks: the blocks of an S 2 (2, 4, 10) which embeds a P(10, 4, 1) (see Section 3) and the following ones 13,5,9], [0, 14,6,10], [1,8,3,13], [1,14,7,4], [1,11,10,5], [1,12,9,6], [4,1,7,11], [5,1,6,12], [9,1,3,14], [10,1,13,8], [2,5,4,3], [2,10,9,11], [2,14,13,12], [2,9,8...…”

“…20. λ = 4, v = 9, w = 4 (note that this is not a minimum embedding, but we need this construction in the proof of Theorem 3.10). Blocks: [3,4,5,6], [6,7,8, 0]; • {a i , x, y, t} for each triple {x, y, t} ∈ R i , where R i , i = 0, 1, 2, 3, are the four parallel classes of a Kirkman triple system on Z 9 ; [13,3,16,9], [15,6,0,10], [13,2,14,7], [7,8,6,1], [6,11,10,13], [11,13,12,6], [11,1,13,0], [16,1,12,14], [12,10,17,14], [0, 8,9,15], [7,4,13,14], [4,…”

“…Necessary conditions for embedding, λ ≥ 3 v (mod 12) v ≥ λ (mod 6) ≥ 3 w 0, 1, 4, 9 4 3 0 3, 7 7 3 1 6, 10 18 for λ = 3, 6 for λ ≥ 9 3 2 0, 3, 6, 9 6 2, 4 1 1, 4, 7, 10 4 2, 4 0 0, 3 When µ = λ = 1, the (minimum) embedding of an H-design into a G-design has been studied for many pairs of graphs H and G with H a subgraph of G [2][3][4]6,[9][10][11][12][14][15][16]. The well-known definition of complementary path decompositions of the complete multigraph can be restated as the minimum embedding of a P(v, 4, λ) into an S 2λ (2, 4, v).…”

“…Base blocks: [∞,1,2,4],[1,4,8,13],[1,7,14, 22],[1,10, 20,9], {1, 2, 7, 12}, {1, 3, 7, 10}. 14. λ = 3, v = 27, w = 1.…”

Minimum embedding of P3‐designs into (K4—e)‐designs

“…Let H ¼ P 3 be the path ½a 1 ; a 2 ; a 3 having vertex set fa 1 ; a 2 ; a 3 g and edges fa 1 ; a 2 g, fa 1 ; a 3 g. The embedding problem of a P 3 -design into a G-design, where G is a graph having 4 nonisolated vertices and 4 edges, has been completely solved [5,10,12,13]. We are interested in embedding a P 3 -design into a ðK 4 À eÞ-design, where K 4 À e is the graph ½a 1 ; a 2 ; a 3 À a 4 having vertex set fa 1 ; a 2 ; a 3 ; a 4 g and edges fa 1 ; a 2 g, fa 1 ; a 3 g, fa 2 ; a 3 g, fa 1 ; a 4 g, and fa 2 ; a 4 g.…”