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arXiv 2107.14767math.CO
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图的区分阈值

Distinguishing threshold of graphs

Mohammad Hadi Shekarriz, Bahman Ahmadi, Seyed Alireza Talebpour Shirazi Fard, Mohammad Hassan Shirdareh Haghighi

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20 pages, 7 figures
发表信息：
Journal of Graph Theory (2022)
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摘要：

图$G$的一个顶点着色称为可区别的，如果$G$的所有非单位自同构都不能保持它，则$G$的可区别数记为$D(G)$，它是实现这种着色所需的最小颜色数，$G$的可区别阈值记为$θ(G)$，它是使$G$的每一个$k$ -着色都可区别的最小数目$k$.作为另一种定义，$θ(G)$比$G$的自同构的循环分解中的最大循环数多一个。在本文中，我们描述$θ(G)$时，$G$是断开的。然后证明了：对于每一个正整数$k\neq 2$，虽然有无穷多个图的判别阈值等于$k$，但我们有$θ(G)=2$当且仅当$\vert V(G)\vert =2$ .证明了若$θ(G)=3$，则要么$G$同构于4个~3顶点图之一，要么是$2p$阶图，其中$p\neq 3,5$是素数.进一步证明了$θ(G)=D(G)$当且仅当$G$是不对称的，$K_n$或$\overline{K_n}$ .最后，我们考虑所有广义Johnson图$J(n,k,i)$，它是$\{1,\ldots , n\}$的所有$k$ -子集上的图，其中两个顶点$A$和$B$相邻，如果$|A\cap B|=k-i$。在研究了它们的自同构群和可区别数之后，我们计算出它们的可区别阈值为$θ(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$，除非是$ k=\frac{n}{2}$和$i\in\{ \frac{k}{2} , k\}$，在这种情况下我们有$θ(J(n,k,i))={n\choose k}$。

英文摘要：

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the distinguishing threshold of $G$, denoted by $θ(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. As an alternative definition, $θ(G)$ is one more than the maximum number of cycles in the cycle decomposition of automorphisms of $G$. In this paper, we characterize $θ(G)$ when $G$ is disconnected. Afterwards, we prove that, although for every positive integer $k\neq 2$ there are infinitely many graphs whose distinguishing thresholds are equal to $k$, we have $θ(G)=2$ if and only if $\vert V(G)\vert =2$. Moreover, we show that if $θ(G)=3$, then either $G$ is isomorphic to one of the four graphs on~3 vertices or it is of order $2p$, where $p\neq 3,5$ is a prime number. Furthermore, we prove that $θ(G)=D(G)$ if and only if $G$ is asymmetric, $K_n$ or $\overline{K_n}$. Finally, we consider all generalized Johnson graphs, $J(n,k,i)$, which are the graphs on all $k$-subsets of $\{1,\ldots , n\}$ where two vertices $A$ and $B$ are adjacent if $|A\cap B|=k-i$. After studying their automorphism groups and distinguishing numbers, we calculate their distinguishing thresholds as $θ(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$, unless $ k=\frac{n}{2}$ and $i\in\{ \frac{k}{2} , k\}$ in which case we have $θ(J(n,k,i))={n\choose k}$.

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