No abstract

Let ( G , + ) be an abelian group and consider a subset A ⊆ G with ∣ A ∣ = k. Given an ordering ( a 1 , … , a k ) of the elements of A, define its partial sums by s 0 = 0 and s j = ∑ i = 1 j a i for 1 ≤ j ≤ k. We consider the following conjecture of Alspach: for any cyclic group Z n and any subset A ⊆ Z n ⧹ { 0 } with s k ≠ 0, it is possible to find an ordering of the elements of A such that no two of its partial sums s i and s j are equal for 0 ≤ i < j ≤ k. We show that Alspach’s Conjecture holds for prime n when k ≥ n − 3 and when k ≤ 10. The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime n a sequence of length k having distinct partial sums exists in any subset of Z n ⧹ { 0 } of size at least 2 k − 8 k in all but at most a bounded number of cases.

No abstract

A subset of an abelian group is sequenceable if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_j$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$.   We demonstrate the sequenceability of subsets of size $k$ of $\mathbb{Z}_n \setminus \{ 0 \}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$.  We obtain similar, but partial, results for $13 \leq k \leq 15$.   This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not  contain $0$ then it is sequenceable.

In 1980, about 20 years after sequenceable groups were introduced by Gordon to construct row-complete latin squares, Keedwell published a survey of all the available results concerning sequencings. This was updated (jointly with Dénes) in 1991 and a short overview, including results about complete mappings and R-sequencings, was given in the CRC Handbook of Combinatorial Designs in 1995. In Sections 1 and 2 we give a survey of the current situation concerning sequencings, including details of the most important constructions. In Section 3 we consider some concepts closely related to sequenceable groups: R-sequencings, harmonious groups, supersequenceable groups (also known as super P-groups), terraces and the Gordon game. We also look at constructions for row-complete latin squares that do not use sequencings.

We use heuristic algorithms to find terraces for small groups. We show that Bailey's Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell's Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups A 6 , S 6 , PSL(2, q 1 ) and PGL(2, q 2 ) where q 1 and q 2 are prime powers with 3 ≤ q 1 ≤ 11 and 3 ≤ q 2 ≤ 8. A sequencing for a group of a given order implies the existence of a complete latin square at that order. We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. This gives 31 new orders at which complete latin squares are now known to exist, the smallest of which is 63. In addition, we consider terraces with some special properties, including constructing a directed T 2 -terrace for the non-abelian group of order 21 and hence a Roman-2 square of order 21 (the first known such square of odd order). Finally we report the total number terraces and directed terraces for groups of order at most 15.MSC: 20D60, 20B15.

a b s t r a c tConstraints are given on graceful labellings for paths that allow them to be used to construct cyclic solutions to the Oberwolfach Problem. We give examples of graceful labellings meeting these constraints and hence, infinitely many new families of cyclic solutions can be constructed.

We use graceful labellings of paths to give a new way of constructing terraces for cyclic groups. These terraces are then used to find cyclic solutions to the three table Oberwolfach problem, ${\rm OP}(r,r,s)$, where two of the tables have equal size. In particular we show that, for every odd $r \geq 3$ and even $r$ with $4 \leq r \leq 16$, there is a number $N_r$ such that there is a cyclic solution to ${\rm OP}(r,r,s)$ whenever $s \geq N_r$. The terraces we are able to construct also prove a conjecture of Anderson: For all $m \geq 3$, there is a terrace of ${\Bbb Z}_{2m}$ which begins $0, 2k, k, \ldots$ for some $k$.