A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $\text{◂()▸}\left(a_1,\text{…},a_k\right)$ of its elements such that the partial sums $\text{◂()▸}\left(s_0,s_1,\text{…},s_k\right)$, given by $s_0=0$ and $s_i=\text{◂∑▸}{\sum }_{j=1}^i{a}_j$ for $1\le i\le k$, are distinct, with the possible exception that we may have $\text{◂=▸}{s}_k=s_0=0$. In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized by Alspach and Liversidge 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 $\text{◂⋅▸}Cay\left[G:\pm A\right]$ such that $\text{◂=▸}\mathrm{\Delta }\left(P\right)=\pm A$. 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 ) and such that $\text{◂=▸}\mathrm{\Delta }\left(W\right)=\pm A$. This is possible given that the partial sums $s_i$ and $s_j$ are different whenever $i$ and $j$ are distinct and $\text{◂≤▸}\mid i-j\mid \le t$. In this case, we say that the set $A$ is $t$-weakly sequenceable. The main result here presented is that any subset $A$ of $\text{◂+▸}{\mathbb{Z}}_p⧹\left\{0\right\}$ is $t$-weakly sequenceable whenever or when $A$ does not contain pairs of type $\left\{x,-x\right\}$ and .

中文翻译：

---

循环群中的弱序列性

一个子集$\mathrm{一个}$阿贝尔群的$G$如果有排序是可排序的$\text{◂()▸}\left({\mathrm{一个}}_1,\text{…},{\mathrm{一个}}_{ķ}\right)$其元素使得部分总和$\text{◂()▸}\left(s_0,s_1,\text{…},s_{ķ}\right)$, 由$s_0=0$和$s_{\mathrm{一世}}=\text{◂∑▸}{\sum }_{j=1}^{\mathrm{一世}}{\mathrm{一个}}_j$为了$1\le \mathrm{一世}\le ķ$, 是不同的, 除了我们可能有的例外$\text{◂=▸}{s}_{ķ}=s_0=0$. 文献中有几个关于阿贝尔群子集可序列性的猜想和问题，Alspach 和 Liversidge 将这些猜想和问题结合起来总结为一个猜想，即如果一个阿贝尔群的子集不包含 0，则它是可序列的。如果可序列集的元素$\mathrm{一个}$总和不为 0 则存在一条简单路径$磷$在凯莱图中$\text{◂⋅▸}C\mathrm{一个}\mathrm{是的}\left[G：\pm \mathrm{一个}\right]$这样$\text{◂=▸}\mathrm{\Delta }\left(磷\right)=\pm \mathrm{一个}$. 在本文中，受这种图论解释的启发，我们提出了对这一猜想的弱化。在这里，在上述假设下，我们想要找到一个排序，其部分和定义了一个步行$W$周长大于$吨$（对于给定的) 并且这样$\text{◂=▸}\mathrm{\Delta }\left(W\right)=\pm \mathrm{一个}$. 这是可能的，因为部分总和$s_{\mathrm{一世}}$和$s_j$无论何时都不一样$\mathrm{一世}$和$j$是不同的并且$\text{◂≤▸}\mid \mathrm{一世}-j\mid \le 吨$. 在这种情况下，我们说集合$\mathrm{一个}$是$吨$-弱序列化。这里提出的主要结果是任何子集$\mathrm{一个}$的$\text{◂+▸}{\mathbb{Z}}_p⧹\left\{0\right\}$是$吨$- 弱序列化或者什么时候$\mathrm{一个}$不包含类型对$\left\{X,-X\right\}$和.