The existence of disjoint (hooked) near‐Rosa sequences and applications

Abstract

We show that the necessary conditions are sufficient for the existence of two disjoint near (hooked) Rosa sequences, with all admissible orders n≥6 and all possible defects. Further, we apply this result for the existence of new types of cyclic and simple GDDs.

Extended near Skolem sequences Part I

“…It is reasonable to anticipate that extended near Skolem sequences will find several new applications. A recent result of Austin and Shalaby [3] contains an application of a restricted class of extended near Skolem sequences. The existence of two disjoint near Rosa sequences (an extended near Skolem sequence with a hook in the middle and a missing difference) [21] was used to produce new constructions of cyclic GDDs with blocks of size 3 and several indices.…”

“…In 1957, Skolem [23], when studying Steiner triple systems, solved the problem of partitioning the numbers n 1, 2, …, 2 into n pairs a b ( , ) r r such that b a r − = r r for r n = 1, 2, …, . For example, for n = 4, the pairs (1, 2), (5,7), (3,6), (4,8) form such a partition. This partition gives rise to the base blocks r b n {0, , + } r , namely, {0, 1, 6}, {0, 2, 11}, {0, 3, 10}, {0, 4, 12}, which when developed modulo 25 yields 100 blocks of a cyclic STS(25).…”