Hooked extended Langford sequences of small and large defects

Abstract

ABSTRACT. It is shown that for m = 2d + 5, 2d + 6, 2d + 7 and d ≥ 1, the set {1, . . . , 2m + 1} − {k} can be partitioned into differencesIt is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, . . . , 2m + 2} − {k, 2m + 1} can be partitioned into differencesThese partitions are used to show that if m ≥ 8d + 3, then the set {1, . . .A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion. The juxtaposition, S 1 S 2 , of sequences S 1 and S 2 is their concatenation. IntroductionFor general information about Langford sequences we refer to Shalaby [12].

Langford sequences and a product of digraphs

“…Skolem, Langford sequences and their many generalizations have applications in numerous areas, see for instance [13]. Although their origin is in the fifties, many recent papers have been contributed to their study and applications from different points of view, as for instance [20,21,23,24]. In this paper, we study Skolem and Langford sequences through (extended) Skolem and Langford labelings of m − → K 2 .…”