On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs

TL;DR

This paper investigates upper bounds for the rectilinear crossing numbers of complete balanced multipartite graphs and layered graphs, contributing to understanding their minimal crossing configurations in plane drawings.

Contribution

It provides new upper bounds on the rectilinear crossing numbers for both complete balanced multipartite and layered graphs.

Findings

Upper bounds established for $K_n^r$ crossing numbers

Upper bounds established for $L_n^r$ crossing numbers

Advances understanding of minimal crossing arrangements

Abstract

A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let $n \ge r$ be positive integers. The graph $K_n^r$, is the complete $r$-partite graph on $n$ vertices, in which every set of the partition has at least $\lfloor n/r \rfloor$ vertices. The layered graph, $L_n^r$, is an $r$-partite graph on $n$ vertices, in which for every $1\le i \le r-1$, all the vertices in the $i$-th partition are adjacent to all the vertices in the $(i+1)$-th partition. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and~$L_n^r$.