Vertex-edge total sum antimagic labeling of graphs

Abstract

Let $G$ be a graph with $p$ vertices and $q$ edges. A vertex--edge total sum antimagic labeling(VETSAML) of $G$ is a bijection $f : V(G) \cup E(G) \to \{1, 2, \dots, p + q\}$ satisfying the following conditions: \begin{enumerate} \item $f(e) \in \{1, 2, \dots, q\}$ for each edge $e \in E(G)$, \item $f(v) \in \{q+1, q+2, \dots, q + p\}$ for each vertex $v \in V(G)$, \item For every vertex $v \in V(G)$, the \emph{total sum} \[ S(v) = f(v) + \sum_{\substack{e \in E(G) \\ e \text{ incident to } v}} f(e) \] \end{enumerate} is distinct among all vertices of $G$. A graph admitting such a labeling is called a vertex--edge total sum antimagic graph. In this paper, we study VETSAML for some several families of graphs, such as cycles, paths, stars, wheels, friendship graphs, and corona-type graphs.