On two conjectures concerning vertex magic total labelings of generalized Petersen graphs

Abstract

A vertex-magic total labeling of a graph with $v$ vertices and $e$ edges is defined as a one-to-one map taking the vertices and edges onto the integers $1,2,\dots ,v+e$ with the property that the sum of the label on a vertex and the labels on its incident edges is a constant, independent of the choice of vertex. In this paper we give a vertex-magic total labeling for the prism $D_n$ for all $n \ge 3$; and a vertex-magic total labeling for the generalized Petersen graphs $P(n,m)$ for $n \ge 3$, $1 \le m \le \lfloor\frac{n-1}{2}\rfloor$, and $n$ and $m$ coprime.