Edge-Locating Coloring of Graphs

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.