Graceful 2-regular graphs and Skolem sequences

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

“…Ì ÓÖ Ñ 2.15º ( [93]) There exists a k-near Rosa sequence of order n if and only if either n ≡ 0, 3 (mod 4) and k is even, or n ≡ 1, 2 (mod 4) and k is odd, with the exceptions when (n, k) = (3, 2), (4,2).…”

“…By considering an exhaustive list of cases and constructing required sequences for each of them, L i n e k and J i a n g [49] proved that these conditions are also sufficient when the defect d is 2 or 3, except for d = 3 and (l, k) = (3, 2), (3,6), (4,1), (4,5), (4,9) when no such sequence exists. In the same paper, they also settled the existence problem of hooked k-extended Langford sequences of defect 2.…”

“…The idea of looped Langford sequences goes back to P r i d a y [74], and it found its application in construction of extended near-Skolem sequences. For example, (2,4,2,7,5,4,8,6,3,5,7,3,0,6,8), (2,4,2,7,5,4,6,8,3,5,7,3,6,0,0,8) is a looped Langford set of defect 2 and order 8.…”

“…Position the first element of RHS at the place of a hook in the HS, and put the last element of the HS at the place of the hook in the RHS. For example, L = (12,10,8,6,4,11,9,7,4,6,8,10,12,5,7,9,11,0,5) is a hooked Langford sequence of defect 4 and length 9. Hooking (1, 1, 2, 0, 2) with L, we get (1, 1, 2, 5, 2, 11, 9, 7, 5, 12, 10, 8,6,4,7,9,11,4,6,8,10,12), which is a 3-near Skolem sequence of order 12 (cf.…”

“…The second definition was studied by N i c k e r s o n [64], and N o w a k o w s k i [66] proved that these two definitions are equivalent. For example, if n = 4, the partition {(2, 3), (6,8), (4,7), (1,5)}, or equivalently, the sequence (4, 1, 1, 3, 4, 2, 3, 2), is a (pure) Skolem sequence of order 4. On the other hand, the partition { (5,6), (1,3), (8,11), (9,13), (2,7), (4, 10)}, or in equivalent notation the sequence (2, 5, 2, 6, 1, 1, 5, 3, 4, 6, 3, 0, 4), is a hooked Skolem sequence of order 6. In order to prove the existence of cyclic Steiner triple systems, R o s a [79] introduced a split Skolem sequence of order n which is an (n + 1)-extended Skolem sequence of order n. Also, a hooked split Skolem sequence of order n is a hooked (n + 1)-extended Skolem sequence of order n. In more recent literature, these sequences are also called Rosa and hooked Rosa sequences of order n.…”