On Rainbow Vertex Antimagic Coloring of Related Prism Graphs and Its Operations

Abstract

Let $G=(V,E)$ be a simple, connected and un-directed graph, for $f:E(G)\rightarrow\{1,2,\dots, |E(G)|\}$, the weight of a vertex $v\in V(G)$ under $f$ is $w_f(v)=\Sigma_{e \in E(v)} f(e)$, where $E(v)$ is the set of vertices incident to $v$. The function $f$ is called vertex antimagic edge labeling if every vertex has distinct weight. While, rainbow vertex coloring is a coloring of graph vertices where each vertex on the graph is connected by a path that all internal vertices on the $u-v$ path have different colors. We introduce a new notion, namely a rainbow vertex antimagic coloring, which is a combination of antimagic labeling and rainbow vertex coloring. The rainbow vertex antimagic connection number of $G$, denoted by $rvac(G)$, is the smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of $G$.  In this paper we aim to discover some new lemmas or theorems regarding to $rvac(G)$.