The Locating Rainbow Connection Number of the Edge Corona of a Graph with a Complete Graph
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arXiv
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| Format: | Preprint |
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2024
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| author | Bustan, Ariestha Widyastuty Salman, A. N. M Putri, Pritta Etriana |
| author_facet | Bustan, Ariestha Widyastuty Salman, A. N. M Putri, Pritta Etriana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09304 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Locating Rainbow Connection Number of the Edge Corona of a Graph with a Complete Graph Bustan, Ariestha Widyastuty Salman, A. N. M Putri, Pritta Etriana Combinatorics 05C78, 05C15 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. |
| title | The Locating Rainbow Connection Number of the Edge Corona of a Graph with a Complete Graph |
| topic | Combinatorics 05C78, 05C15 |
| url | https://arxiv.org/abs/2410.09304 |