On 1-vertex bimagic vertex labeling

Abstract

A 1-vertex magic vertex labeling of a graph $G$ with $p$ vertices is defined as a bijection $f$ from the vertices to the integers $1, 2, \ldots, p$ with the property that there is a constant $k$ such that at any vertex $x$, $\sum_{y \in N(x)} f(y) = k$, where $N(x)$ is the set of vertices adjacent to $x$. In this paper we introduce 1-vertex bimagic vertex labeling of a graph $G$ and obtain the necessary condition for a graph to be 1-vertex bimagic. We exhibit the same type of labeling for some class of graphs and give some general results.