Weak sequenceability in cyclic groups

A subset $A$ of an abelian group G $G$ is sequenceable if there ordering ( a 1 , … k ) $({a}_{1},\ldots ,{a}_{k})$ its elements such that the partial sums s 0 $({s}_{0},{s}_{1},\ldots ,{s}_{k})$ given by = ${s}_{0}=0$ and i ∑ j ${s}_{i}={\sum }_{j=1}^{i}{a}_{j}$ for ≤ $1\le i\le k$ are distinct, with possible exception we may have ${s}_{k}={s}_{0}=0$ . In literature several conjectures questions concerning sequenceability subsets groups, which been combined summarized Alspach Liversidge into conjecture does not contain then it sequenceable. If set do sum to exists simple path P $P$ in Cayley graph C y [ : ± ] $Cay[G:\pm A]$ Δ ${\rm{\Delta }}(P)=\pm A$ this paper, inspired graph–theoretical interpretation, propose weakening conjecture. Here, under above assumptions, want find whose define walk W $W$ girth bigger than t $t$ (for < $t\lt }}(W)=\pm This ${s}_{i}$ ${s}_{j}$ different whenever $i$ $j$ distinct ∣ − $| i-j| \le t$ case, say -weakly The main result here presented any Z p ⧹ { } ${{\mathbb{Z}}}_{p}\setminus \{0\}$ 7 7$ or when pairs type x $\{x,-x\}$ 8 8$