Construction of locally plane graphs with many edges

TL;DR

This paper introduces a method to construct $k$-locally plane graphs with more than linear edges, using randomized thinning procedures for edge-colored bipartite graphs, advancing understanding of geometric graph density.

Contribution

It presents a novel construction of $k$-locally plane graphs with super-linear edges and develops randomized thinning techniques applicable to bipartite graphs.

Findings

Constructed $k$-locally plane graphs with super-linear edges.

Developed randomized thinning procedures for edge-colored bipartite graphs.

Applicable techniques for other graph problems.

Abstract

A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most $k$ in a geometric graph $G$ is self-intersecting we call $G$ $k$-locally plane. The main result of this paper is a construction of $k$-locally plane graphs with a super-linear number of edges. For the proof we develop randomized thinning procedures for edge-colored bipartite (abstract) graphs that can be applied to other problems as well.