Infinite Trees with Finite Dimensions

Abstract

: The properties of graph we consider are metric dimension, partition dimension, and locating-chromatic number. Infinite graphs can have either infinite or finite dimension. Some necessary conditions for an infinite graph with finite metric dimension has been studied in 2012. Infinite graphs with finite metric dimension will also have finite partition dimension and locating-chromatic number. In this paper we find a relation between the partition dimension (locating chromatic number) of an infinite tree with the metric dimensions of its special subtree. We also show that it is possible for an infinite trees with infinite metric dimension to have finite partition dimension (locating-chromatic number).