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

Daniel Gonçalves

TL;DR

This work proves Scheinerman's conjecture for the 3-colorable case by showing every 3-colorable planar graph admits a 3-slopes segment representation. The authors introduce Triangular 3-slopes Contact representations (TC-representations) and TC-schemes built on Eulerian triangulations, and encode the construction with a linear system $\mathcal{L}$ on face-sizes $x_f$. They prove $\mathcal{L}$ has a unique solution using a biadjacency-matrix argument based on perfect matchings, and show that such a solution yields a TC-scheme from which a 3-slopes segment representation follows. Consequently, parallel segments align with color classes, establishing a PURE-$3$-DIR representation for 3-colorable planar graphs, though the jump to four slopes is ruled out by counterexamples, leaving open questions for higher slopes. The results advance understanding of intersection representations for planar graphs and connect combinatorial, linear-algebraic, and geometric perspectives on triangulations.

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.

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

TL;DR

This work proves Scheinerman's conjecture for the 3-colorable case by showing every 3-colorable planar graph admits a 3-slopes segment representation. The authors introduce Triangular 3-slopes Contact representations (TC-representations) and TC-schemes built on Eulerian triangulations, and encode the construction with a linear system on face-sizes . They prove has a unique solution using a biadjacency-matrix argument based on perfect matchings, and show that such a solution yields a TC-scheme from which a 3-slopes segment representation follows. Consequently, parallel segments align with color classes, establishing a PURE--DIR representation for 3-colorable planar graphs, though the jump to four slopes is ruled out by counterexamples, leaving open questions for higher slopes. The results advance understanding of intersection representations for planar graphs and connect combinatorial, linear-algebraic, and geometric perspectives on triangulations.

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.

Paper Structure

This paper contains 22 sections, 7 theorems, 12 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Terminology
  3. TC-representations and TC-schemes
  4. (1) $\Longrightarrow$ (2).
  5. (2) $\Longrightarrow$ (1).
  6. The linear system model
  7. $\mathop{\mathrm{\mathcal{L}}}\nolimits$ has a unique solution
  8. A solution of $\mathop{\mathrm{\mathcal{L}}}\nolimits$ defines a TC-scheme
  9. If every face $f\in F_1$ is such that $x_f\neq 0$,
  10. (Step 1)
  11. (Step 2)
  12. If some faces $f\in F_1$ are such that $x_f= 0$,
  13. If $a'$ has only zero valued incident faces in $T$,
  14. If $a_i$ has only zero valued incident faces in $T$,
  15. If both $a_i$ and $a'$ have non-zero valued incident faces in $T$,
  16. ...and 7 more sections

Key Result

Theorem 1

Every 3-colored planar graph has a 3-slopes segment representation such that parallel segments correspond to the color classes.

Figures (17)

  • Figure 1: The octahedron and a 3-slopes contact representation. It is unique, up to vertex automorphism, up to scaling, and once the slopes are set.
  • Figure 2: (left) Vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ (middle) A TC-representation $\mathop{\mathrm{\mathcal{R}}}\nolimits$ with various types of intersection points. The circled ones correspond to particular intersection points with more than three segments. (right) Its corner graph $C(\mathop{\mathrm{\mathcal{R}}}\nolimits)$, where gray faces correspond to the degenerate faces (i.e. they correspond to intersection points in $\mathop{\mathrm{\mathcal{R}}}\nolimits$). The dark gray faces are particular degenerate faces. One has size six, and there are two faces of size three that correspond to the same intersection point.
  • Figure 3: From left to right. A TC-representation $\mathop{\mathrm{\mathcal{R}}}\nolimits$, with the size of its inner triangles; its corner graph $C(\mathop{\mathrm{\mathcal{R}}}\nolimits)$, where gray faces are the degenerate faces; and two triangulations where $\mathop{\mathrm{\mathcal{R}}}\nolimits$ is a TC-scheme of both.
  • Figure 4: (left) The size of the triangles around $a_0$. (right) The size of the triangles around some inner vertex $b_i$.
  • Figure 5: (a) Example of a Eulerian triangulation $T$ (dashed lines), with incidence graph $I$. The numbers correspond to the solution of $\mathop{\mathrm{\mathcal{L}}}\nolimits$. (b) The graph $I'$ obtained after (Step 1). (C) The graph $G^\Delta$.
  • ...and 12 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Lemma 8
  • Claim 9
  • Claim 10
  • ...and 2 more