Weak Sequenceability in Cyclic Groups

TL;DR

This paper introduces the concept of t-weak sequenceability in abelian groups, exploring conditions under which subsets can be ordered to produce walks with large girth in Cayley graphs, extending classical sequenceability conjectures.

Contribution

It proposes a weakening of the sequenceability conjecture, defining t-weak sequenceability, and proves that subsets of cyclic groups are t-weak sequenceable for t<7 or t<8 under certain conditions.

Findings

Any subset of Z_p extbackslash extbraceleft 0 extbraceright is t-weak sequenceable for t<7.

If the subset does not contain pairs of the form extbackslash{}{x, -x extbackslash{}}, it is t-weak sequenceable for t<8.

The concept extends classical sequenceability by allowing partial sums to form walks with girth larger than t.

Abstract

A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_i$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $s_k = s_0 = 0$. In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized in $[4]$ into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set $A$ do not sum to $0$ then there exists a simple path $P$ in the Cayley graph $Cay[G:\pm A]$ such that $\Delta(P) = \pm A$. In this paper, inspired by this graph-theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk $W$ of girth bigger than $t$ (for a given $t < k$) and such that $\Delta(W) = \pm A$. This is possible given that the partial sums $s_i$ and $s_j$ are different whenever $i$ and $j$ are distinct and $|i-j|\leq t$. In this case, we say that the set $A$ is $t$-weak sequenceable. The main result here presented is that any subset $A$ of $\mathbb{Z}_p\setminus \{0\}$ is $t$-weak sequenceable whenever $t<7$ or when $A$ does not contain pairs of type $\{x,-x\}$ and $t<8$.