论文标题
循环组的弱测序性
Weak Sequenceability in Cyclic Groups
论文作者
论文摘要
如果有$（a_1，\ ldots，a_k）$的子集$ a $ abelian $ g $是可以进行测序的\ leq k $是不同的，除了我们可能具有$ s_k = s_0 = 0 $的例外。在文献中，有几个关于阿贝尔群体子集测序性的猜想和问题，这些猜测已在$ [4] $中组合并总结在猜想中，即如果阿贝尔组的子集不包含0，则可以进行测序。如果可测序集$ a $的元素不贵于$ 0 $，则存在一个简单的路径$ p $中的cayley图$ cay $ cay [g：\ pm a] $，因此$δ（p）= \ pm a $。 在本文的启发下，我们提出了这种猜想的削弱。在这里，根据上述假设，我们希望找到一个订购的订单，其部分款项定义了大于$ t $（给定的$ t <k $）的$ w $ w $ a Walk $ w $（对于给定的$ t <k $），并且$δ（w）= \ pm a $。鉴于部分总和$ s_i $和$ s_j $在$ i $和$ j $不同，$ | i-j | \ leq t $时都不同。在这种情况下，我们说$ a $ a $是$ t $ weak-weak可测序。这里提出的主要结果是，任何子集$ a $ a $ a $ a $ \ m artbb {z} _p \ setMinus \ {0 \} $是$ t $ - weak-weak序列可用$ t <7 $或$ a $ a $不包含$ \ \ \ \ \ \ {x，-x，-x，-x \} $和$ t <8 $的$ t <7 $。
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 $Δ(P) = \pm A$. 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 $Δ(W) = \pm A$. 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 $t<8$.
