Vertex-magic total labelings of disconnected graphs

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 $\lambda$.

In this paper, we present the vertex magic total labelings of disconnected graph, in particular, two copies of isomorphic generalized Petersen graphs $2P(n,m)$, disjoint union of two non-isomorphic suns $S_m \cup S_{n}$ and $t$ copies of isomorphic suns $tS_n$.