Trees with certain locating-chromatic number

Abstract

The locating-chromatic number of a graph G can be defined as 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 in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of υ to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n − 1. They also proved that there exists a tree of order n, n ≥ 5, having locating-chromatic number k if and only if k ϵ {3,4,…, n − 2, n}. In this paper, we characterize all trees of order n with locating-chromatic number n − t, for any integers n and t, where n > t + 3 and 2 ≤ t