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 v 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 ∈{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 < n /2.