arXiv 1807.083342018-07math.COcs.DM

图的度量维数与模式避免

Metric dimension and pattern avoidance in graphs

Jesse Geneson

摘要：

本文证明了度量维数或边度量维数有界的图的模式避免的若干结果。我们证明了直径$D$和边度量维数$k$的图中的最大可能边数至多为$(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i)^{k-1}$，从而锐化了Zubrilina（2018）的$\binom{k}{2}+k D^{k-1}+D^{k}$的界。我们还证明了当度量维数为$\leq k$的图包含完全图$K_{n}$作为子图时，$n$的最大值是$n = 2^{k}$ .证明了当度量维数为$\leq k$的图包含完全二部图$K_{n,n}$作为子图时，$n$的最大值为$2^{Θ(k)}$ .进一步证明了边度量维数为$\leq k$的图含有$K_{1,n}$作为子图时，$n$的最大值为$n = 2^{k}$ .我们还证明了当度量维数$\leq k$的图包含$K_{1,n}$作为子图时，$n$的最大值是$3^{k}-O(k)$ .此外，我们还证明了$d$维网格$\prod_{i = 1}^{d} P_{r_{i}}$的边度量维数至多为$d$ .这推广了Kelenc等人（2016）的两个结果，即非路径网格具有边度量维数$2$，以及$d$维超立方体具有至多$d$的边度量维数。我们还提供了一个表征$n$ -顶点图的边度量尺寸$n-2$，回答一个问题Zubrilina。作为这一特征的结果，我们证明了任何连通$n$ -顶点图$G$，使$edim(G) = n-2$有直径至多$5$。更一般地，我们证明了任何连通$n$ -顶点图的边度量维数$n-k$有直径最多$3k-1$。

英文摘要：

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$ is at most $(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i)^{k-1}$, sharpening the bound of $\binom{k}{2}+k D^{k-1}+D^{k}$ from Zubrilina (2018). We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete graph $K_{n}$ as a subgraph is $n = 2^{k}$. We prove that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete bipartite graph $K_{n,n}$ as a subgraph is $2^{Θ(k)}$. Furthermore, we show that the maximum value of $n$ for which some graph of edge metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $n = 2^{k}$. We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $3^{k}-O(k)$. In addition, we prove that the $d$-dimensional grids $\prod_{i = 1}^{d} P_{r_{i}}$ have edge metric dimension at most $d$. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension $2$ and that $d$-dimensional hypercubes have edge metric dimension at most $d$. We also provide a characterization of $n$-vertex graphs with edge metric dimension $n-2$, answering a question of Zubrilina. As a result of this characterization, we prove that any connected $n$-vertex graph $G$ such that $edim(G) = n-2$ has diameter at most $5$. More generally, we prove that any connected $n$-vertex graph with edge metric dimension $n-k$ has diameter at most $3k-1$.

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