The Locating Rainbow Connection Number of the Edge Corona of a Graph with a Complete Graph

TL;DR

This paper investigates the locating rainbow connection number of graphs formed by the edge corona of a graph with a complete graph, establishing tight bounds for this parameter.

Contribution

It determines the upper and lower bounds of the locating rainbow connection number for edge corona graphs involving complete graphs, proving these bounds are tight.

Findings

Established tight bounds for the locating rainbow connection number.

Analyzed the edge corona of a graph with a complete graph.

Provided theoretical results on graph coloring parameters.

Abstract

A graph has a locating rainbow coloring if every pair of its vertices can be connected by a path passing through internal vertices with distinct colors and every vertex generates a unique rainbow code. The minimum number of colors needed for a graph to have a locating rainbow coloring is referred to as the locating rainbow connection number of a graph. Let $G$ and $H$ be two connected, simple, and undirected graphs on disjoint sets of $|V(G)|$ and $|V(H)|$ vertices, $|E(G)|$ and $|E(G)|$ edges, respectively. For $j\in\{1,2,...,|E(G_m)|\}$, the edge corona of $G_m$ and $H_n$, denoted as $G_m \diamond H_n$, is constructed by using a single copy of $G_m$ and $E(G_m)$ copies of $H_n$, and then connecting the two end vertices of the $j$-th edge of $G_m$ to every vertex in the $j$-th copy of $H_n$. In this paper, we determine the upper and lower bounds of the locating rainbow connection number for the class of graphs resulting from the edge corona of a graph with a complete graph. Furthermore, we demonstrate that these upper and lower bounds are tight.