油漆成本与节约判别数

摘要：

你会得到一个顶点为中性色的图，并被要求用$d$颜色中昂贵的颜料为顶点的子集着色，这样只有平凡的对称性才能保留颜色类。您的目标是最小化需要这种昂贵绘制的顶点数。这篇文章讨论了围绕你的选择的问题。特别是，一个图被称为$d$ -可区分的，如果存在一个着色与$d$颜色，使得只有平凡的自同构保持颜色类。$G$的可区分数，表示为${\rm Dist}(G)$，是$G$是$d$可区分的最小$d$。我们定义$d$ -distinguishing的-paint成本，表示为$ρ^d(G)$，是需要绘制到$d$ -distinguish $G$的顶点的最小数量。该费用因$d$而异。$G$的最大油漆成本称为上油漆成本，表示为$ρ^u(G)$，并在$d={\rm Dist}(G)$时出现;最小油漆成本称为下油漆成本，表示为$ρ^\ell(G)$。此外，我们定义最小的$d$，其油漆成本是$ρ^\ell(G)$，是节俭的区别数，${\rm Fdist}(G)$。在本文中，我们正式定义$ρ^d(G)$，$ρ^u(G)$，$ρ^\ell(G)$和${\rm Fdist}(G)$。我们还表明，$ρ^u(G)$和$ρ^\ell(G)$，以及${\rm Fdist}(G)$和${\rm Dist}(G)$，可以是任意大的倍数彼此。最后，我们找到书图$B_{m,n}$的这些参数，总结如下。对于$n\geq 2$和$m\geq 4$，我们显示$\bullet$ $ρ^\ell(B_{m,n}) = n-1;$ $\bullet$ $ρ^u(B_{m,n}) \geq (m-2) \left( n-k^{m-3} \right) +1$，其中$k={\rm Dist}(B_{m,n});$ $\bullet$ ${\rm Fdist}(B_{m,n}) = 2+\left\lfloor \frac{n-1}{m-2} \right\rfloor.$

英文摘要：

You are handed a graph with vertices in a neutral color and asked to color a subset of vertices with expensive paints in $d$ colors in such a way that only the trivial symmetry preserves the color classes. Your goal is to minimize the number of vertices needing this expensive paint. This paper address the issues surrounding your choices. In particular, a graph is said to be $d$-distinguishable if there exists a coloring with $d$ colors so that only the trivial automorphism preserves the color classes. The distinguishing number of $G$, denoted ${\rm Dist}(G)$, is the smallest $d$ for which $G$ is $d$-distinguishable. We define the -paint cost of $d$-distinguishing, denoted $ρ^d(G)$, to be the minimum number of vertices that need to be painted to $d$-distinguish $G$. This cost varies with $d$. The maximum paint cost for $G$ is called the upper paint cost, denoted $ρ^u(G)$, and occurs when $d={\rm Dist}(G)$; the minimum paint cost is called the lower paint cost, denoted $ρ^\ell(G)$. Further, we define the smallest $d$ for which the paint cost is $ρ^\ell(G)$, to be the frugal distinguishing number, ${\rm Fdist}(G)$. In this paper we formally define $ρ^d(G)$, $ρ^u(G)$, $ρ^\ell(G)$, and ${\rm Fdist}(G)$. We also show that $ρ^u(G)$ and $ρ^\ell(G)$, as well as ${\rm Fdist}(G)$ and ${\rm Dist}(G)$, can be arbitrarily large multiples of each other. Lastly, we find these parameters for the book graph $B_{m,n}$, summarized as follows. For $n\geq 2$ and $m\geq 4$, we show $\bullet$ $ρ^\ell(B_{m,n}) = n-1;$ $\bullet$ $ρ^u(B_{m,n}) \geq (m-2) \left( n-k^{m-3} \right) +1$, where $k={\rm Dist}(B_{m,n});$ $\bullet$ ${\rm Fdist}(B_{m,n}) = 2+\left\lfloor \frac{n-1}{m-2} \right\rfloor.$

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