On the Biplanar Crossing Number of Kn

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the Euclidean plane. The k-planar crossing number crk(G) of G is min{cr(G1) + cr(G2) + . . .+ cr(Gk)}, where the minimum is taken over all possible decompositions of G into k subgraphs G1, G2, . . . , Gk. The problem of computing the crossing number of complete graphs, cr(Kn), exactly for small n and bounding its value for large n has been the subject of extensive recent research. In this paper we examine the biplanar crossing number of complete graphs, cr2(Kn). Since 1971, Owens’ construction [IEEE Transactions on Circuit Theory, 18(2):277– 280, 1971] has been the best known construction for biplanar drawings of Kn for large values of n. We propose an improved technique for constructing biplanar drawings of Kn, which reduces the lower order terms of Owens’ upper bound. For small fixed n, we show that cr2(K10) = 2, cr2(K11) ∈ {4, 5, 6}, and for n ≥ 12, we improve previous upper and lower bounds on cr2(Kn).