Locating-chromatic number of the edge-amalgamation of trees

Dian Kastika Syofyan, Edy Tri Baskoro, Hilda Assiyatun

Abstract


The investigation on the locating-chromatic number of a graph was initiated by Char- trand 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.


Keywords


edge-amalgamation; locating-chromatic number; terminal edge; tree

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DOI: http://dx.doi.org/10.19184/ijc.2020.4.2.6

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