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

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.

Figures (6)

• Figure 1: A locating rainbow coloring of $G_m \diamond K_n$
• Figure 2: A rainbow vertex coloring of (a) $T_7$ and (b) $T_7 \diamond K_3$
• Figure 3: Graph $P_m \diamond K_n$
• Figure 4: Graph $C_m \diamond K_n$
• Figure 8: Graf $K_m \diamond K_n$.

Theorems & Definitions (24)

• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7