Perfect Systems Of Difference Sets Research Articles

Regular Perfect Systems of Sets of Iterated Differences

Fors≥2, a set {a(i,j):1 ≤j≤s+1 −i≤s} wherea(1,j), 1≤ j≤s, are some prescribed integers anda(i+1,j) =|a(i,j) −a(i,j+1)|, for 1≤i<sand 1≤j≤s−i, is called a set of iterated differences. Such a set has sizesand is full if it containss(s+1)/2 distinct integers. Kreweras and Loeb suggested the problem of partitioning a run ofms(s+1)/2 integers starting withcintomfull sets of iterated differences of sizes. We show that necessary conditions for this are that 2≤s≤9, and thatmbe sufficiently large in comparison withc. In particular, a single set of iterated differences of sizescontains the integers 1 tos(s+1)/2 (inclusive) iff 2 ≤s≤5. We also discuss connections between this problem and the theory of perfect systems of difference sets.

Uniform perfect systems of sets of iterated differences of size 4 1

A general problem raised by the work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into sets, known as sets of iterated differences, the s( s + 1)/2 elements of which can be expressed as certain linear combinations of s integer valued parameters. In this paper we study the case of four parameters. By examining properties already present in examples with only a single set, we build up the rudiments of an arithmetic of these partitions comparable to that for perfect systems of difference sets and complete permutations.

On critical perfect systems of difference sets

A perfect system of difference sets with threshold c is a partition of a consecutive run of integers beginning with c into full difference sets of valency at least 2. The BKT inequality, due to Bermond, Kotzig and Turgeon gives a necessary condition for the existence of such systems; systems for which this inequality holds with equality are called critical. We show that a critical perfect system of difference sets with threshold c which contains no difference sets of valency 2 consists of 2 c - 1 difference sets of valency 3. We also discuss, in the light of this result, other inequalities for perfect systems which are stronger than the BKT inequality at least in some circumstances.

On the BKT inequality and improved bounds for perfect systems of difference sets

We present a generalization of the BKT Inequality, an inequality for perfect systems of difference sets due to Bermond, Kotzig, and Turgeon, and deduce from this generalization bounds for perfect systems which in some cases improve on those provided by the BKT Inequality. We discuss, in particular, the so-called critical case in which the BKT Inequality holds with equality.

Improved bounds for perfect systems of difference sets

We give some improvements on the BKT inequality, an inequality for perfect systems of difference sets due to Bermond, Kotzig, and Turgeon, and consider, in particular, their implications for the critical case in which the BKT inequality holds with equality.

Further results on irregular, critical perfect systems of difference sets II: systems without splits

An ( m, n; u, v; c)-system is a collection of components, m of valency u−1 and n of valency v−1, whose difference sets form a perfect system with threshold c. If there is an ( m, n; 3, 6; c)-system, then m⩾2 c−1; and if there is a (2 c−1, n; 3, 6; c)-system, then 2 c−1⩾ n. For all sufficiently large c, there are (2 c−1, n; 3, 6; c)-systems with a split at 3 c+6 n−1 at least when n=1, 5, 6 and 7, but such systems do not exist for n=2, 3 or 4. We describe here a general method of construction for (2 c−1, n; 3, 6; c)-systems and use it to show that there are such systems for 2⩽ n⩽4 and certain values of c depending on n. We also discuss the limitations of this method.

Further results on irregular, critical perfect systems of difference sets I: split systems

An ( m, n;u, v;c)-system is a collection of components, m of valency u – 1 and n of valency v − 1, whose difference sets form a perfect system with threshold c. If there is an ( m, n;3, 6; c)-system, then m ⩾ 2 c − 1; and if there is a (2 c − 1, n; 3, 6; c)-system, then 2 c − 1 ⩾ n. For all sufficiently large c, there are (2 c − 1, n; 3, 6; c)-systems at least when n = 1 or 2 and, in particular, (2 c minus; 1, 1; 3, 6; c)-systems which have a certain splitting property enabling them to be pulled apart nicely. We show here that if, for some c and n, there is a (2 c − 1, n; 3, 6; c)-system which splits at 3 c + 6 n − 1, then, in the first place, c − 1 ⩾ n, and, secondly, there is a (2 c ∗ − 1, n; 3, 6, c ∗)- system with a split at 3 c ∗ + 6 n − 1 for all sufficiently large c ∗ depending on c and n. We then confirm the existence of such split systems at least when n = 1, 5, 6 and 7, finding also that they do not exist for n = 2, 3 or 4. We discuss the bearing of these results on the study of critical perfect systems and on the multiplication theorem for these systems. Another approach to (2 c − 1, n; 3, 6, c)-systems, including the cases n = 2, 3 and 4, is considered in the sequel.

CRITICAL PERFECT SYSTEMS OF DIFFERENCE SETS WITHOUT COMPONENTS OF SIZE TWO

Journal Article CRITICAL PERFECT SYSTEMS OF DIFFERENCE SETS WITHOUT COMPONENTS OF SIZE TWO Get access D. G. ROGERS D. G. ROGERS Department of Mathematical Sciences The UniversityAberdeen AB9 2TY Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 43, Issue 2, June 1992, Pages 253–258, https://doi.org/10.1093/qmath/43.2.253 Published: 01 June 1992 Article history Received: 15 October 1989 Published: 01 June 1992

On the General Erdös Conjecture for Perfect Systems of Difference Sets and Embedding Partial Complete Permutations

We confirm the (general) Erdös conjecture for perfect systems of difference sets; namely that, for each threshold c ⩾ 1, all sufficiently long finite runs of consecutive integers beginning with c can be partitioned into such systems. We use this result to show further that, for each c ⩾ 1, the set of integers n with c ⩽ ∣ n ∣ < c + l admits complete permutations for all sufficiently large l . Consequently, if the spectra of certain kinds of constraints for complete permutations are not empty then they contain all sufficiently large odd positive integers. Our concern in this paper is only with qualitative results, and not with the question of how large is sufficiently large in each of these three assertions.

Irregular, extremal perfect systems of difference sets II

An (m, n; u, v; c)-system is a collection of components,m of valencyu ? 1 andn of valencyv ? 1, whose difference sets form a perfect system with thresholdc. A necessary condition for the existence of an (m, n; u, v; c)-system foru = 3 or 4 is thatm ? 2c ? 1; and there are (2c ? 1,n; 3, 6;c)-systems for all sufficiently largec at least whenn = 1 or 2. It is shown here that if there is a (2c ? 1,n; u, 6;c)-system thenn = 0 whenu = 4 andn ≤ 2c ? 1 whenu = 3. Moreover, if there is a (2c ? 1,n; 3, 6;c)-system with a certain splitting property thenn ≤ c ? 1, this last result being of possible interest in connection with the multiplication theorem for perfect systems.

An arithmetic of complete permutations with constraints, I: an exposition of the general theory

We develop an arithmetic of complete permutations of symmetric, integral bases; this arithmetic is comparable to that of perfect systems of difference sets with which there are several interrelations. Super-position of permutations provides the addition of this arithmetic. Addition if facilitated by complete permutations with a certain “splitting” property, allowing them to be pulled apart and reassembled. The split permutations also provide a singular direct product for complete permutations in conjunction with the multiplication (direct product) of the arithmetic which itself derives from that for perfect systems of difference sets. We pay special attention to complete permutations satisfying constraints both fixed and variable; this is equivalent to embedding partial complete permutations in complete permutations. In the sequel, using this arithmetic, we investigate the spectra of certain constraints with respect to central integral bases which are of interest for the purpose of giving further constructions either of complete permutations with constraints or of irregular, extremel perfect systems of difference sets.

An arithmetic of complete permutations with constraints, II: case studies

Using the arithmetic of complete permutations developed in the first part of this paper, we investigate the spectra of certain constraints with respect to central, integral bases which are of interest for the purposes of giving further constructions either of complete permutations with constraints or of irregular, critical perfect systems of difference sets. We also present, in the appendices, catalogues of examples and enumerative data based on computer studies.

A note on perfect systems of difference sets

Etude des systemes parfaits densembles de differences. Raccourcissement des formules donnees par P.J. Laufer et J.M. Turgeon. Extension a d'autres valeurs de e

Compatible additive permutations of finite integral bases

Let? be an additive permutation of a finite integral base. It is shown that ifB is symmetric, then there is a unique additive permutation? ofB which is compatible with? in the sense that?? ?1 is also an additive permutation; and that, further, ifB is asymmetric, then there is no additive permutation ofB which is compatible with?. Thus, in the symmetric case, there are no additively compatible sets (of permutations) forB of size greater than 3. This contrasts with the situation for completely compatible sets (equivalently, additive sequences of permutations) where for certainB compatible sets of size (resp. length) 4 or less are known, but where nothing is known of sets of greater size (resp. length). It is also noted how this result restricts the possibility of a useful multiplication theorem for the additive analogue of perfect systems of difference sets and graceful graphs.

The minimum number of components in 4-regular perfect systems of difference sets

The number of components m in regular ( m, 5, c)-systems is given in the literature to date by the inequality m ⩾ 4 c − 2 ( Bermond et al., “Proceedings, 18th Hungarian Combin. Colloq.”, North-Holland, Amsterdam, 1976 ). The case m = 4 c − 2 is called extremal. It is proved that (4 c − 2, 5, c)-systems do not exist. An example of a (4 c, 5, c)-system with c = 2, is given. Since, in a 4-regular system, m must be even, loc. cit., it is concluded that the lower bound on the number of components is given by m >/ 4 c.

Inequalities for perfect systems of difference sets

Inequalities for perfect systems of difference sets

On a conjecture of Paul Erdös for perfect systems of difference sets

Let m, n 1, n 2,…, n m and c be positive integers. Let A = { A 1, A 2, … A m} be a system of sequences of integers A i = ( a i1 < a 12 < … < a in i ; i=1,…, m, and let D i = { a ij − a ik ‖ 1 ⩽ k < j ⩽ n i } be the difference set of A j . Then S = { D 2, D 2,…, D m} is a perfect system of difference sets if D=D 1⋃D 2⋃⋯⋃D m= c,c+1,…,c−1+ ∑ i=1 m n i 2 Such a system is trivial if n i = 2 for at least one i. Paul Erdös conjectured that, for every positive integer e, except for a finite number of them, there is a non-trivial perfect system of difference sets whose differences are the first e positive integers. These exceptions are discussed and a proof of the conjecture is given.

Addition Theorems for Perfect Systems of Difference Sets

Two addition theorems are proved for perfect systems of difference sets. In both, the addition depends on the existence of such systems with a certain ‘splitting’ property which allows them to be pulled apart and reassembled. We obtain further evidence in support of Bermond's conjecture that (m, 4, l)-systems exist for m ⩾ 4.

Regular perfect systems of difference sets

Regular perfect systems of difference sets