Locating-chromatic number of the edge-amalgamation of trees

Abstract

The investigation on the locating-chromatic number of a graph was initiated by Chartrand et al. (2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph G as the smallest integer k such that there exists a k-partition of the vertex-set of G such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For i = 1, 2, . . . , t, let Ti be a tree with a fixed edge eoi called the terminal edge. The edge-amalgamation of all Tis denoted by Edge-Amal{Ti;eoi} is a tree formed by taking all the Tis and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.