Ioan Tomescu and Muhammad Imran: On Metric and Partition Dimensions of Some Infinite Regular Graphs, p.461-472

Abstract:

In this paper some infinite regular graphs generated by tilings of the plane by regular triangles and hexagons are considered. These graphs have no finite metric bases but their partition dimension is finite and is evaluated in some cases. Also, it is proved that for every $n\geq 2$ there exists finite induced subgraphs of these graphs having metric dimension equal to $n$ as well as infinite induced subgraphs with metric dimension equal to three.

Key Words: Metric dimension, partition dimension, plane tiling, infinite regular graph, induced subgraph.

2000 Mathematics Subject Classification: Primary: 05C12,
Secondary: 05C35.

Download the paper in pdf format here.