A Short Proof of the Planarity Characterization of Colin de Verdière

Colin de Verdiere introduced an interesting new invariant ?(G) for graphs G, based on algebraic and analytic properties of matrices associated with G. He showed that the invariant is monotone under taking miners and moreover, that ?(G) ? 3 if only if G is planar. In this paper we give a short proof of Colin de Verdiere?s result that ?(G) ? 3 if G is planar.