On rainbow vertex anti-magic coloring of amalgamation graphs

A rainbow vertex anti-magic coloring represents a relatively new area of exploration in graph theory. This concept extends the idea of rainbow vertex coloring by incorporating elements of anti-magic labeling. Given a function f : E(G)→ {1, 2, …,|E(G)|}, the weight of a vertex v ∈ V(G) under f is given by wf (v) = Σe∈E(v) f(e), where E(v) denotes the set of edges incident to v. The function f is classified as a vertex anti-magic edge labeling if the weights assigned to all vertices are distinct. A path is referred to as a rainbow path if for any pair of vertices u and v all internal vertices along the u – v path possess distinct weights. The rainbow vertex anti-magic connection number of G, denoted as rvac(G), is the minimum number of colors required across all rainbow colorings derived from a rainbow vertex anti-magic labeling of G. This paper presents the computation of the rainbow vertex anti-magic connection number for specific graph families, including the Dutch windmill, diamond graph, octopus graph, and amalgamation graph.