Crossing numbers of complete tripartite and balanced complete multipartite graphs

TL;DR

This paper establishes new bounds for the rectilinear and general crossing numbers of complete tripartite and balanced complete multipartite graphs, using flag algebras and asymptotic analysis to improve understanding of their crossing number behavior.

Contribution

It introduces an analogous bound for the crossing number of tripartite graphs and analyzes the asymptotic crossing number limits for balanced complete multipartite graphs.

Findings

Proved that cr'(K_{n_1,n_2,n_3}) (n_1,n_2,n_3) for large n.

Established lower bounds: 0.973 A(n,n,n) r'(K_{n,n,n}) and 0.666 A(n,n,n) r(K_{n,n,n}).

Defined z(r) as an upper bound for the limit superior of the crossing number ratio.

Abstract

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n,n,n}) and 0.666 A(n,n,n)\le cr(K_{n,n,n}), with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r-partite graph. Richter and Thomassen proved in 1997 that the limit as n\to\infty of cr(K_{n,n}) over the maximum number of crossings in a drawing of K_{n,n} exists and is at most 1/4. We define z(r)=3(r^2-r)/8(r^2+r-3) and show that for a fixed r and the balanced complete r-partite graph, z(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.