Bounds and Methods for k-Planar Crossing Numbers

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K_{2k+1,q}, for k >= 2. We prove tight bounds for complete graphs