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 \( ν_k(G)\), 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 \( K_n\). We determine \( ν_k(K_n)\) whenever \( 2 < n/k ≤ 3\) and improve the lower bounds on \( ν_k(K_n)\) 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