Not all planar graphs are in PURE-4-DIR

Daniel Gonçalves

doi:10.7155/jgaa.00533

Abstract

We prove that some planar graphs are not intersection graphs of segments if only four slopes are allowed for the segments, and if parallel segments do not intersect. This refutes a conjecture of D. West [D. West, SIAM J. Discrete Math. Newsletter, 1991].

Highlights

The intersection graphs of segments class SEG is a widely studied class of graphs
GH does not admit a PURE-4-DIR representation. It remains open whether plane graphs belong to PURE-k-DIR, for some constant k > 4
N vertices belongs to PURE-(n − f (n))-DIR, for some function f tending to infinity. Another question is whether plane graphs belong to k-DIR, for some k > 0

Summary

Introduction

The intersection graphs of segments class SEG is a widely studied class of graphs. Any graph G ∈ SEG admits a representation in the plane, where each vertex v ∈ V (G) corresponds to a segment v, and where two segments u and v intersect if and only if uv ∈ E(G). A graph belongs to k-DIR if it has an intersection representation with segments using at most k different slopes. Let us introduce the class PURE-k-DIR of graphs having a pure k-DIR representation, that is where parallel segments do not intersect. He proved that outerplanar graphs belong to PURE-3-DIR and he conjectured that every planar graph belongs to SEG. West conjectured a strengthening of Scheinerman’s conjecture [14] He asked whether every planar graph has a PURE-4-DIR representation. In a pure representation parallel segments form a stable set This conjecture would strengthen the 4-color theorem. We prove Theorem 1 in Section 3 using signed planar graphs that are not 4-colorable [10] (in the sense of signed graphs)

Planar gadget

Proof of Theorem 1

Conclusion