The existence of near-Skolem and hooked near-Skolem sequences

Abstract

Let m, n be integers, m <n. A near-Skolem sequence of order n and defect m is a sequence S=(s1,s2, . . . . s,,_,)ofintegers.~~{1,2 )..., m-l,m+l, . . . . n) whichsatisfies the following conditions: (1) For every ke{1,2 ,..., m-l,m+l,..., n) there are exactly two elements si, Sj~S, such that si = sj = k. (2) If si=sj=k then j-i=k. A hooked near-Skolem sequence of order n and defect m is a sequence HS = (si , s2 ,. . . , szn _ 1) of integers Sip{ 1,2, . . . , m 1, m + 1, . . . , n} satisfying conditions (1) (2) and the condition: (3) s+z=o. We will refer to near-Skolem and hooked near-Skolem sequences of order n and defect m as m-near-Skolem and hooked m-near-Skolem sequences, respectively. Skolem (1957) [9] showed that the set (1,. . . , 2n) can be partitioned into distinct pairs (a,, b,) such that b,-a, =I, r = 1, . . _, n. He proved that the necessary and sufficient condition for the existence of such partition is n = 0,l (mod 4). O’Keefe (1961) [7] proved that the necessary and sufficient condition for partitioning the set (1, . . . ,2n 1,2n + 1) into distinct pairs (a,, b,) such that b,a,=r, r= 1, . . ., n, is n=2,3 (mod4). The results of Skolem and O’Keefe led to the construction of cyclic Steiner triple systems of order v E 1 (mod 6). Note that if we did not skip the difference m in the above definitions, we would obtain the definitions of Skolem and hooked Skolem sequences, respectively. The pairs of subscripts (i, j) whenever si = sj= k give the required partitioning in both Skolem and O’Keefe cases.