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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.

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w(u)\ne w(v)$, where $w(u) = f(u) + \sum_{e\in E(u)} f(e)$ and $E(u)$ is the set of incident edge(s) of $u$. The local antimagic total chromatic number, denoted $\chi_{lat}(G)$, is the minimum number of distinct weights over local antimagic total labeling of $G$. In this paper, we provide a correct proof and exact local antimagic total chromatic number of path and spider graphs given in [Local vertex antimagic total coloring of path graph and amalgamation of path, {\it CGANT J. Maths Appln.} {\bf 5(1)} 2024, DOI:10.25037/cgantjma.v5i1.109]. Further, we determined the local antimagic total chromatic number of spider graph with each leg of length at most 2. We also showed the existence of unicyclic and bicyclic graphs with local antimagic total chromatic number 3.

The locating rainbow connection number of a graph is defined as the minimum number of colors required to color vertices such that every two vertices there exists a rainbow vertex path and every vertex has a distinct rainbow code. This rainbow code signifies a distance between vertices within a given set of colors in a graph. This paper aims to determine the locating rainbow connection number for vertex-transitive graphs. Three main theorems are derived, focusing on the locating rainbow connection number for some vertex-transitive graphs, Comment: The article omprises a total of 11 pages and includes a total of 9 figures