Note on k-planar crossing numbers

The crossing numberCR(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k-planar crossing number of G, CRk(G), is defined as the minimum of CR(G0)+CR(G1)+…+CR(Gk−1) over all graphs G0,G1,…,Gk−1 with ∪i=0k−1Gi=G. It is shown that for every k≥1, we have CRk(G)≤(2k2−1k3)CR(G). This bound does not remain true if we replace the constant 2k2−1k3 by any number smaller than 1k2. Some of the results extend to the rectilinear variants of the k-planar crossing number.