有界度量维数图的极值结果
Extremal results for graphs of bounded metric dimension
摘要：
度量维数是一个图形参数，它是由机器人导航、药物设计和图像处理中的问题所激发的。在本文中，我们回答了（Geneson，度量维数和模式避免，离散应用数学284，2020，1-7）中关于图的度量维数和模式避免的几个开放极值问题。具体地说，我们构造了一类新的图族，它允许我们确定度量维数至多为$k$的图的最大可能度，度量维数至多为$k$的图的最大可能退化度，度量维数至多为$k$的图的最大可能色数，以及最大$n$，对于该最大$n$，存在度量维数至多为$k$的图，该图包含$K_{n, n}$。我们还调查了一个变种的度量维度称为边缘度量维度和解决另一个问题，从同一纸$n$足够大，通过显示，边缘度量维度的$P_n^{d}$是$d$的$n \geq d^{d-1}$。此外，我们使用的概率参数取得进展的另一个开放的问题，从同一文件显示，最大可能的团数的边缘度量维数$k$的图是$2^{Θ(k)}$。我们还在（N）的一个问题上取得了进展。Zubrilina，On the edge dimension of a graph，Discrete Math.341，2018，2083-2088）通过找到一族新的三元组$(x, y, n)$，对于该三元组，存在度量维度$x$、边缘度量维度$y$和阶数$n$的图。特别是，我们发现，对于每个整数$k > 0$，存在图$G$与度量维度$k$，边度量维度$3^k(1-o(1))$，和顺序$3^k(1+o(1))$。
英文摘要：
Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most $k$, the maximum possible degeneracy of a graph of metric dimension at most $k$, the maximum possible chromatic number of a graph of metric dimension at most $k$, and the maximum $n$ for which there exists a graph of metric dimension at most $k$ that contains $K_{n, n}$. We also investigate a variant of metric dimension called edge metric dimension and solve another problem from the same paper for $n$ sufficiently large by showing that the edge metric dimension of $P_n^{d}$ is $d$ for $n \geq d^{d-1}$. In addition, we use a probabilistic argument to make progress on another open problem from the same paper by showing that the maximum possible clique number of a graph of edge metric dimension at most $k$ is $2^{Θ(k)}$. We also make progress on a problem from (N. Zubrilina, On the edge dimension of a graph, Discrete Math. 341, 2018, 2083-2088) by finding a family of new triples $(x, y, n)$ for which there exists a graph of metric dimension $x$, edge metric dimension $y$, and order $n$. In particular, we show that for each integer $k > 0$, there exist graphs $G$ with metric dimension $k$, edge metric dimension $3^k(1-o(1))$, and order $3^k(1+o(1))$.
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