Edge-Locating Coloring of Graphs

TL;DR

This paper introduces the concept of edge-locating coloring in graphs, determines exact values for specific graphs, explores relationships with graph parameters, and characterizes this coloring for trees and other graph classes.

Contribution

The paper initiates the study of edge-locating coloring, providing exact values, characterizations, and bounds for various classes of graphs, including trees, complete graphs, and join graphs.

Findings

Exact values of (G) for some graphs

Characterization of graphs with (G) in  and 

Relationship between (G+K_1) and (G)

Abstract

An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits an edge-locating coloring is the edge-locating chromatic number of $G$, and denoted by $\chi'_L(G)$. In this paper we initiate to introduce the concept of edge-locating coloring and determine the exact values $\chi'_L(G)$ of some custom graphs. The graphs $G$ with $\chi'_L(G)\in \{2,m\}$ are characterized, where $m$ is the size of $G$. We investigate the relationship between order, diameter, and edge-locating chromatic number of $G$. For a complete graph $K_n$, we obtain the exact values of $\chi'_L(K_n)$ and $\chi'_L(K_n-M)$, where $M$ is a maximum matching; indeed this result is also extended for any graph. We will determine the edge-locating chromatic number of join graph $G+H$, where $G$ and $H$ are some well-known graphs. In particular, for any graph $G$, we show a relationship between $\chi'_L(G+K_1)$ and $\Delta(G)$. We investigate the edge-locating chromatic number of trees and present a characterization bound for any tree in terms of maximum degree, number of leaves, and the support vertices of trees. Finally, we prove that any edge-locating coloring of a graph is an edge distinguishing coloring.