On dimensions of some infinite regular graphs generated by infinite hexagonal grid

Abstract

The partition dimension and metric dimension of a connected graph are related as pd(G) ≤ dim(G) + 1. However, the partition dimension may be much smaller than the metric dimension and this phenomena is called a discrepancy between metric dimension and partition dimension [12].In this paper some infinite regular graphs generated by tilings of the plane by regular hexagons are considered. These graphs have no finite metric bases but their partition dimension is finite and is evaluated in some cases.It is natural to ask for the characterization of graphs having discrepancies between their metric and partition dimension.