Irregular Extremal Perfect Systems of Difference Sets

An {m, n;u, u;c)-system is a perfect system of difference sets with m components of size u— 1 and n components of size v— 1 having threshold c. A necessary condition for the existence of an (m, n; u, 6; c)system for u = 3 or 4 is that m ^ 2c— 1. We show that, in the extremal case when m = 2c— 1, there are (2c— 1, n; 3, 6; c)-systems for c ^ 3 when n = 1, and for c ^ 5 when n = 2, with a few possible exceptions, by first characterizing the components of size 5 and then drawing on results on complete permutations with constraints to construct the components of size 3. Large classes of non-extremal (m, n; 3, 6; c)-systems can be constructed from these extremal families; and the thrust of our work is to suggest that the necessary condition that m > 2c— 1 is also sufficient when c is sufficiently large compared with n. 1. Perfect systems of difference sets The difference set D(A) of a set A = {at: 1 ^ / < w} of integers at, 1 ̂ / ^ M, is the set If the integers \ai — a^, 1 ^ i <j ^u, are all distinct, so that D(A) contains I j distinct integers, then D(A) is said to be full. For a positive integer c, a system of difference sets which is perfect for c, in the sense of [3], or, as is sometimes said, a perfect system of difference sets with threshold c is a partition of a run of consecutive integers starting with c into full difference sets. These systems are the subject of the survey [1] which contains much information and many further references. In the case when c = 1, perfect systems of difference sets are equivalent to graceful valuations of windmill graphs; see [1, pp. 7-9] and, for further details and references on graceful graphs [2]. To be more precise in the specification of these systems, let u = (« l 5 . . . , um) be an w-tuple of positive integers ur, 1 ^ r ^ m, and let Ar = {a(r, i):\ ^ / < ur). Then the sets Dr = D(Ar), 1 ^ r ^ m, form a perfect system of difference sets with threshold cif (J Dr = {i:c^i<c+d(m;u)}, (1) r-i where The underlying sets Ar, 1 < r < m are then said to form an (m;u;c)-system with components Ar, 1 ^ r ^ m. The size of component Ar, as also of the difference set Dr, is ur— 1; and, to avoid triviality, we suppose that ur ^ 3, 1 < r ^ m. If all the Received 4 December 1984. 1980 Mathematics Subject Classification 05B10. J. London Math. Soc. (2) 34 (1986) 193-211 7 JLM 34