### Edge-locating coloring of graphs

#### 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 *χ*′_{L}(*G*). This paper introduces and studies the concept of edge-locating coloring. Graphs *G* with *χ*′_{L}(*G*)∈{2, *m*} are characterized, where *m* is the size of *G*. We investigate the relationship between order, diameter and edge-locating chromatic number. We obtain the exact values of *χ*′_{L}(*K*_{n}) and *χ*′_{L}(*K*_{n} − *M*), where *M* is a maximum matching; indeed this result is also extended for any graph. We determine the edge-locating chromatic number of the join graphs of some well-known graphs. In particular, for any graph *G*, we show a relationship between *χ*′_{L}(*G* + *K*_{1}) and *Δ*(*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.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2024.12.1.6

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