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

TL;DR

This paper improves bounds on the k-page book crossing number of complete graphs and determines exact values for certain ratios, using bounds on convex graphs with small local crossing numbers.

Contribution

It advances the understanding of crossing numbers in k-page book drawings and characterizes maximum edges in convex graphs with limited local crossing numbers.

Findings

Improved lower bounds on ν_k(K_n) for all k ≥ 14.

Exact values of ν_k(K_n) for 2 < n/k ≤ 3.

Determined maximum edges in convex graphs with local crossing number ≤ 4.

Abstract

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 the graph $ G $, denoted by $ \nu_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ \nu_k(K_n) $ for all $ k\geq 14 $ and determine $ \nu_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.