论文标题
具有任意程度的整数距离挖掘家族的统治比率
Domination ratio of a family of integer distance digraphs with arbitrary degree
论文作者
论文摘要
整数距离挖掘是所有整数的加法组$ \ mathbb {z} $的cayley图$γ（\ mathbb {z}，s）$，相对于有限的子集$ s \ subseteq \ subseteq \ mathbb {z} $。 $γ（\ mathbb {z}，s）$的支配比（定义为其主导集的最小密度）与某些数字理论问题有关，例如铺平整数并找到一组缺失差异的整数的最大密度。我们精确地确定了整数$ d $ d $ d $ d $ d $ d $ d $ d \ d \ ge2 $和$ d \ ge2 $和$ s \ notin [0，d-2] $的整数距离图$γ（\ Mathbb {z}，\ {1,2，\ ldots，d-rdots，d-ldots，d-ldots，d-ldots，d-d-2，s \} $。我们的结果概括了图$γ（\ Mathbb {z}，\ {1，s \}）$的统治比，并在\ Mathbb {z} \ setMinus \ {0,1 \} $中，也意味着某些COIDMANTH $ abb $ gumbb s $γ（z）$γ（Z）$γ（\ math）的统治数字，也意味着$ \ mathbb {z} _n $是整数的有限循环群，modulo $ n $，$ s $是$ \ mathbb {z} _n $的子集。
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $Γ(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$. Our result generalizes a previous result on the domination ratio of the graph $Γ(\mathbb{Z},\{1,s\})$ with $s\in\mathbb{Z}\setminus\{0,1\}$ and also implies the domination number of certain circulant graphs $Γ(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the finite cyclic group of integers modulo $n$ and $S$ is a subset of $\mathbb{Z}_n$.
