New Bounds on k-Planar Crossing Numbers
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The crossing number cr(G) of a graph G is the minimum number of crossings over all possible drawings of G in the plane. Analogously, the k-planar crossing number of G, denoted by cr_k(G), is the minimum number of crossings over all possible drawings of the edges of G in k disjoint planes. We present new bounds on the k-planar crossing number of complete graphs and complete bipartite graphs. In particular, for the case of k=2, we improve the current best lower bounds on biplanar crossing numbers by a factor of 1.37 for complete graphs, and by a factor of 1.34 for complete bipartite graphs. We extend our results to the k-planar crossing number of complete (bipartite) graphs, for any positive integer k ≥ 2. To better understand the relation between crossing numbers and biplanar crossing numbers, we pose a new problem of finding the largest crossing number that implies biplanarity. In particular, we prove that for every graph G, cr(G) ≤ 10 implies cr_2(G)=0.
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