Vertex-magic total labeling of the union of suns

Abstract

Let $G$ be a graph with vertex set $V=V(G)$ and edge set $E=E(G)$ and let $e=\vert E(G) \vert$ and $v=\vert V(G) \vert$. A one-to-one map $\lambda$ from $V\cup E$ onto the integers $\{ 1,2, ..., v+e \}$ is called {\it vertex magic total labeling} if there is a constant $k$ so that for every vertex $x$, \[ \lambda (x) \ +\ \sum \lambda (xy)\ =\ k \] where the sum is over all vertices $y$ adjacent to $x$. Let us call the sum of labels at vertex $x$ the {\it weight} $w_{\lambda}(x)$ of the vertex under labeling $\lambda$; we require $w_{\lambda}(x)=k$ for all $x$. The constant $k$ is called the {\it magic constant} for $\lambda$.

A sun $S_n$ is a cycle on $n$ vertices $C_n$, for $n \ge 3$, with an edge terminating in a vertex of degree 1 attached to each vertex.

In this paper, we present the vertex magic total labeling of the union of suns, including the union of $m$ non-isomorphic suns for any positive integer $m \ge 3$, proving the conjecture given in [6].