On RAC Drawings of Graphs with one Bend per Edge

TL;DR

This paper investigates the maximum number of edges in 1-bend RAC graphs, establishing tighter bounds and demonstrating the existence of graphs with nearly the maximum possible edges, advancing understanding of graph density with right-angle crossings.

Contribution

It improves the bounds on the maximum edge count of 1-bend RAC graphs, narrowing the gap between known upper and lower limits.

Findings

Maximum edges in 1-bend RAC graphs are at most 5.5n-O(1).

Existence of infinitely many 1-bend RAC graphs with 5n-O(1) edges.

Previous bounds were 6.5n-O(1) (upper) and 4.5n-O(√n)) (lower).

Abstract

A k-bend right-angle-crossing drawing or (k-bend RAC drawing}, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph that admits a k-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of 1-bend RAC graphs. We show that an n-vertex 1-bend RAC graph cannot have more than $5.5n-O(1)$ edges. We also demonstrate that there exist infinitely many n-vertex 1-bend RAC graphs with exactly $5n-O(1)$ edges. Our results improve both the previously known best upper bound of $6.5n-O(1)$ edges and the corresponding lower bound of $4.5n-O(\sqrt{n})$ edges by Arikushi et al. (Comput. Geom. 45(4), 169--177 (2012)).