On partitions of {1,…,2m+ 1}\{k} into differencesd,…,d+m− 1: Extended Langford sequences of large defect

Abstract

It is shown that for m = 2d − 1, 2d, 2d + 1, and d ≥ 1, the set {1, 2,…, 2m + 2}, − {2,k} can be partitioned into differences d,d + 1,…,d + m − 1 whenever (m,k) ≡ (0,0), (1,d + 1), (2, 1), (3,d) (mod (4,2)) and (d,m,k) ≠ (1,1,3), (2,3,7) (where (x,y) ≡ (u,ν) mod (m,n) iff x ≡ u (mod m) and y ≡ ν (mod n)). It is also shown that if m ≥ 2d − 1 and m ∉ [2d + 2, 8d − 5], then the set {1, 2, …, 2m + 1} − {k} can be partitioned into differences d,d + 1,…,d + m − 1 whenever (m,k) ≡ (0, 1), (1,d), (2,0), (3,d + 1) mod (4,2). Finally, for d = 4 we obtain a complete result for when {1,…,2m + 1} − {k} can be partitioned into differences 4,5,…,m + 3. © 2004 Wiley Periodicals, Inc.

Semi-cyclic holey group divisible designs with block size three

“…We have that if p satisfies p − 5 √ p − 12 > 0, which yields p > 45, then there exists an element x ∈ Z p such that x ∈ C 2 1 , x + 1 ∈ C 2 1 and x − 1 ∈ C 2 0 . When 5 ≤ p ≤ 43, it is readily checked that we may take x as (p, x) = (5, 2), (7,5), (11,6), (13,5), (17,5), (19,2), (23,10), (29,2), (31,11), (37,5), (41, 6), (43,2). ✷ Lemma 6.9 There exists a 3-SCHGDD of type (6, 2 p ) for any prime p ≥ 3.…”

A survey of Skolem-type sequences and Rosa’s use of them

“…[53]). More generalizations of Skolem sequences were needed for the construction of some combinatorial designs.…”

“…Now we look into a technique called pivoting of an element, which is very useful in constructions of extended sequences. We will illustrate it on an example from a construction in [53]. The sequence (6,7,3,4,5,3,6,4,7,5) is a Langford sequence of defect 3 and length 5.…”

Computational Graph Theory

“…The cases 6t + 17 ≤ n ≤ 6t + 20 and the cases (n, t) ∈ {(28, 1), (52,5), (70,8), (72,8)} are dealt with first. Since h ≥ 2, it follows from 3t + 4q + h = n−1 2 − 6 that in each of these cases we have q = 0.…”

“…Since 3t + 4q + h = n−1 2 − 6, and since we have already dealt with the cases where (n, t) ∈ {(27, 1), (28, 1), (52,5), (70,8), (72,8)}, this leaves us with the cases where (n, t, h) in {(27, 1, 4), (29,1,5), (30,1,5), (51,5,4), (53,5,5), (54,5,5), (69,8,4), (71,8,5)}.…”