On Book Crossing Numbers of the Complete Graph

. A \ (k\) -page book drawing of a graph \ (G\) is a drawing of \ (G\) on \ (k\) halfplanes with a line \ (l\) as a common boundary such that the vertices are located on \ (l\) and the edges cannot cross \ (l\). The \ (k\) -page book crossing number of the graph \ (G\), denoted by \ (ₖ (G) \), is the minimum number of edge-crossings over all \ (k\) -page book drawings of \ (G\). This paper improves previous results on \ (k\) -page book crossing numbers of the complete graph \ (Kₙ\). We determine \ (ₖ (Kₙ) \) whenever \ (2 n/k 3\) and improve the lower bounds on \ (ₖ (Kₙ) \) for all \ (k 14\). Our proofs rely on bounding the number of edges in convex geometric graphs with few crossings per edge. Keywordscrossing numberlocal crossing numberbook crossing numberbook drawingscomplete graphconvex geometric graphMSC codes05C1005C3552C10