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Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs

Oswin Aichholzer, Joachim Orthaber, Birgit Vogtenhuber

Abstract

It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each pair of vertices" and show that this stronger conjecture holds for several classes of simple drawings, including strongly c-monotone drawings and cylindrical drawings. As a second main contribution, we give an overview on different classes of simple drawings and investigate inclusion relations between them up to weak isomorphism.

Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs

Abstract

It is a longstanding conjecture that every simple drawing of a complete graph on vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each pair of vertices" and show that this stronger conjecture holds for several classes of simple drawings, including strongly c-monotone drawings and cylindrical drawings. As a second main contribution, we give an overview on different classes of simple drawings and investigate inclusion relations between them up to weak isomorphism.
Paper Structure (23 sections, 18 theorems, 18 figures)

This paper contains 23 sections, 18 theorems, 18 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Outline.
  4. An overview on classes of simple drawings
  5. Straight-line drawings.
  6. $X$-monotone and $x$-bounded drawings.
  7. $2$-page-book drawings.
  8. Shellable drawings.
  9. C-monotone drawings.
  10. Strongly c-monotone drawings.
  11. Cylindrical drawings.
  12. Strongly cylindrical drawings.
  13. Crossing maximal drawings.
  14. Twisted and generalized twisted drawings.
  15. Generalized convex drawings.
  16. ...and 8 more sections

Key Result

Lemma 4

Let $\mathcal{D}$ be a c-monotone drawing of $K_{n}$. Then the following are equivalent:

Figures (18)

  • Figure 1: (aeppv-2017-ssdcge-2017-rrs) A star-simple drawing that has no crossing-free Hamiltonian cycle.
  • Figure 2: (a) An $x$-monotone drawing of $K_{6}$. (b) A crossing minimal $2$-page-book drawing of $K_{8}$. The spine is drawn dashed purple, the completely uncrossed Hamiltonian cycle is drawn orange.
  • Figure 3: (a) The wedge $\Lambda_{e}$ (shaded seagreen) of the lightblue edge $e$. The darkblue edge $f$ together with $e$ covers the plane, which is not allowed in a strongly c-monotone drawing. (b) Illustration of \ref{['lem:strong-c-mon-for-kn']}: A pair of non-incident edges (blue) covering the plane enforces a pair of incident edges covering the plane; either with the darker ($g_{1}$) or with the lighter ($g_{2}$) version of $g$. (c) Condition 3 of \ref{['lem:strong-c-mon-for-kn']}: The star is not crossed by any ray within the seagreen wedge.
  • Figure 4: (a) Hill's drawing $\mathcal{Z}_{9}$ drawn in the plane. The two concentric circles are drawn violet, the rim edges are seagreen, all other circle edges are orange, and the lateral edges are lightblue. (b) An arbitrary (not strongly) cylindrical drawing of $K_{9}$ with the same color coding.
  • Figure 5: The five non-isomorphic simple drawings of $K_{5}$: (a) The convex straight-line drawing $\mathcal{C}_{5}$. (a) -- (c) The three types of straight-line drawings of $K_{5}$. (c) In some sense Hill's drawing $\mathcal{Z}_{5}$. (e) The twisted drawing $\mathcal{T}_{5}$. (a) and (e) The two crossing maximal drawings of $K_{5}$.
  • ...and 13 more figures

Theorems & Definitions (20)

  • Conjecture 1: Rafla r-1988-gdcg
  • Conjecture 2
  • Lemma 4
  • Lemma 10
  • Corollary 11
  • Theorem 12
  • Corollary 13
  • Proposition 15
  • Lemma 16
  • Corollary 17
  • ...and 10 more