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A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $halfplanes with common boundary $ l $, a line, where the vertices are on $ l $and the edges cannot cross $ l $. The $ k $-page book crossing number of thegraph $ G $, denoted by $ \nu_k(G) $, is the minimum number of edge-crossingsover all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graphon $n$ vertices. We improve the lower bounds on $ \nu_k(K_n) $ for all $ k\geq14 $ and determine $ \nu_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs relyon bounding the number of edges in convex graphs with small local crossingnumbers. In particular, we determine the maximum number of edges that a convexgraph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.

Book crossing numbers of the complete graph and small local convex crossing numbers