Neighbor-locating colorings in graphs

Abstract

A k-coloring of a graph G is a k-partition Π = {S 1 , . . . , S k } 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 χ 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.Henning and O. R. Oellermann introduced the so-called metric-locating-dominating sets, by merging the concepts of metric-locating set and dominating set.In [14], G. Chartrand, E. Salehi and P. Zhang, brought the notion of metric location to the ambit of vertex partitions, introducing the resolving partitions, also called metriclocating partition, and defining the partition dimension. Metric location and domination, in the context of vertex partitions, are studied in [28]. In [11], there were introduced the so-called locating colorings considering resolving partitions formed by independents sets.Neighbor location in sets was introduced by P. Slater in [39]. Given a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S is adjacent to some vertex in S. A set S ⊆ V (G) is a locating-dominating set if S is a dominating set and N (u) ∩ S = N (v) ∩ S for every two different vertices u and v not in S. The location-domination number of G, denoted by λ(G), is the minimum cardinality of a locating-dominating set. In [8,27], bounds for this parameter are given. In this paper, merging the concepts studied in [11,39], we introduce the neighbor-locating colorings and the neighbor-locating chromatic number, and examine this parameter in some families of graphs.

The Neighbor-Locating-Chromatic Number of Pseudotrees

“…For P n with 3 ≤ n ≤ 9 and for C n with n ∈ {3, 5, 7, 9}, a 3-NL-coloring is displayed in Figure 1. Clearly, χ N L (C 4 ) = 4 since, as was proved in [1], for every complete multipartite graph G of order n, χ N L (G) = n.…”

“…Both neighbor-locating colorings and the neighbor-locating chromatic number of a graph were introduced in [1]. As a straightforward consequence of these definitions the following remark is derived.…”

“…In [1], we started the study of neighbor-locating partitions formed by independent sets, which we named neighbor-locating colorings. More specifically, we considered vertex colorings such that any two vertices with the same color can be distinguished from each other by the colors of their respective neighbors.…”

“…Remove the edge joining the vertex u of degree 2 and color 2 with the vertex v of degree 3 and color k − 1. Attach a leaf colored with k − 1 to vertex u and a leaf colored with 2 to vertex v. We obtain a tree T k of order 1 2 k(k − 1)(k + 2) − 4 + 2 = 1 2 k(k − 1)(k + 2) − 2 (see an example in Figure 7). We claim that in such a way we have a k-NL-coloring of the tree T k .…”

Neighbor-locating coloring: graph operations and extremal cardinalities

“…In [2], bounds of this parameter are given. In [1], merging the concepts of papers [3,8], there were introduced the so-called neighbor-locating colorings and the neighborlocating chromatic number, studying this parameter in some basic families of graphs.…”