  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 $t<8$.

Combinatorics