Notes on large angle crossing graphs

TL;DR

This paper investigates the properties of a-angle crossing graphs, generalizing the concept of right angle crossing graphs, and provides bounds on their maximum number of edges for all angles between 0 and Pi/2.

Contribution

It extends the study of crossing graphs by establishing bounds on the number of edges in a-angle crossing graphs for all crossing angles a.

Findings

Established upper and lower bounds for edges in aAC graphs for all 0 < a < Pi/2.

Generalized the concept of RAC graphs to a-angle crossing graphs.

Connected the properties of aAC graphs with known results for RAC graphs.

Abstract

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2.