Neighbor-locating colorings in graphs

Abstract

A k-coloring of a graph G is a k-partition II = { S1 , … , Sk } of V (G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u , v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number X N L (G) is the minimum cardinality of a neighbor-locating coloring of G. We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n ≥ 5 with neighbor-locating chromatic number n or n − 1 . We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.