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 'L(G). In this paper we initiate to introduce the concept of edge-locating coloring and determine the exact values 'L(G) of some custom graphs. The graphs G with '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 of G. For a complete graph Kn, we obtain the exact values of 'L(Kn) and 'L(Kn-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 'L(G+K1) and (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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