3-colorable planar graphs have an intersection segment representation using 3 slopes

TL;DR

This paper proves Scheinerman's conjecture that 3-colorable planar graphs can be represented as intersection graphs of line segments with only three slopes, using an approach related to contact representations of planar graphs.

Contribution

It establishes that 3-colorable planar graphs admit intersection segment representations with only three slopes, confirming a longstanding conjecture.

Findings

3-colorable planar graphs can be represented with 3 slopes

The proof adapts methods from contact representations of planar graphs

Supports the conjecture by Scheinerman on slope-restricted representations

Abstract

In his PhD Thesis, E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L. Isenmann, and C. Pennarun. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.