Extremal results for graphs of bounded metric dimension

Abstract: 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 $2^{\Theta(k)}$. 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 $3^k(1+o(1))$.