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Holes in Convex and Simple Drawings

Helena Bergold, Joachim Orthaber, Manfred Scheucher, Felix Schröder

Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex~drawings.

Holes in Convex and Simple Drawings

Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of -holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex~drawings.
Paper Structure (7 sections, 9 theorems, 1 equation, 7 figures)

This paper contains 7 sections, 9 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Holes in convex drawings
  3. Generalized Holes
  4. Properties of vertices inside a k-gon
  5. Proof of Proposition 4.5
  6. Proof of Lemma 2.3
  7. Simple drawings with quadratically many empty 4-cycles

Key Result

Theorem 1

For every sufficiently large $n$, every convex drawing of $K_n$ contains a 6-hole.

Figures (7)

  • Figure 1: A drawing of \ref{['fig:convex']} an $n$-gon $\mathcal{C}_n$ and \ref{['fig:twisted']} a twisted $\mathcal{T}_n$ for $n \ge 4$.
  • Figure 3: The drawing $\mathcal{T}_n'$ without empty 4-triangulations for $n \ge 6$.
  • Figure 4: Constructing the drawing $\mathcal{D}_n$ of $K_n$, $n$ odd, with few empty 4-cycles from $K_5$.
  • Figure 5: A $k$-gon with an interior vertex $v$ in the convex side of the triangle $v_i,v_{i+1},v_{i+2}$.
  • Figure 6: Illustration of the proof of \ref{['lemma:4holes']}.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1: Empty Hexagon theorem for convex drawings
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Proposition 5
  • Theorem 6
  • Conjecture 7: BFRS2024
  • Corollary 8
  • Lemma 9: ArroyoMRS2022
  • Lemma 10