Neighbor-Locating Colorings in Graphs

TL;DR

This paper introduces the neighbor-locating chromatic number, a new graph coloring parameter, and establishes bounds, characterizations, and behavior under graph operations for various graph classes.

Contribution

It defines the neighbor-locating chromatic number, provides tight bounds, characterizes extremal graphs, and analyzes its behavior under join, union, and specific graph families.

Findings

Established tight bounds in terms of graph parameters.

Characterized all graphs with maximum neighbor-locating chromatic number.

Analyzed the parameter for join, union, split, and Mycielski graphs.

Abstract

A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{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 $\chi _{_{NL}}(G)$ 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 $n\geq 5$ 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.