Calculating an upper bound of the locating-chromatic number of trees

Abstract

The locating-chromatic number of a graph $G(V,E)$ is the cardinality of a minimum resolving partition of the vertex set $V(G)$ such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices not contained in the same partition class. Determining the locating-chromatic number of any tree is a difficult task. In this paper, we propose an algorithm to compute the upper bound on the locating-chromatic number of any tree. To do so, we decompose a tree into caterpillars and then compute the upper bound of the locating-chromatic number of this tree in terms of the ones for these caterpillars.