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Plane Hamiltonian Cycles in Convex Drawings

Helena Bergold, Stefan Felsner, Meghana M. Reddy, Joachim Orthaber, Manfred Scheucher

Abstract

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph $K_n$ admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length $Ω(\log n / \log \log n)$ and of a plane matching of size $Ω(n^{1/2})$ in every simple drawing of $K_{n}$. Instead of studying simpler substructures, we prove Rafla's conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of $K_n$ contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane $k$-cycle for each $3 \leq k \leq n$ (pancyclicity), and present further results on maximal plane subdrawings.

Plane Hamiltonian Cycles in Convex Drawings

Abstract

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length and of a plane matching of size in every simple drawing of . Instead of studying simpler substructures, we prove Rafla's conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane -cycle for each (pancyclicity), and present further results on maximal plane subdrawings.
Paper Structure (7 sections, 14 theorems, 10 figures)

This paper contains 7 sections, 14 theorems, 10 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Hamiltonian paths and cycles
  4. Maximal plane subdrawings
  5. Stars and cycles
  6. Proof of Theorem \ref{['theorem:convex_HC']} (Sketch)
  7. Conclusion

Key Result

Lemma 3

Let $b$ be a bad edge and let $e$ be a non-star edge connecting two vertices from the non-convex side of $T_b$. Then $e$ does not cross the triangle $T_b$ and is therefore contained in the non-convex side of $T_b$.

Figures (10)

  • Figure 1: In a simple drawing, edges are not allowed to \ref{['fig:simple_obstructions_selfint']} cross themselves or \ref{['fig:simple_obstructions_through-vertex']} pass through vertices. \ref{['fig:simple_obstructions_touching']} If two edges meet in their relative interior, they have to cross (no touchings). \ref{['fig:simple_obstructions_doublecross']} Each pair of edges crosses at most once and \ref{['fig:simple_obstructions_adjcross']} adjacent edges do not cross.
  • Figure 2: The five non-isomorphic ways to draw the complete graph $K_5$. Type IV and type V are non-convex because the red triangles have no convex side, as witnessed by the blue edges.
  • Figure 3: A bad edge $b = \{v,v+1\}$ with witness $w$ (blue). The triangle $T_b$ is highlighted in red.
  • Figure 6: Constructing a plane Hamiltonian cycle in a convex drawing.
  • Figure 7: \ref{['fig:noprescribededges_a']} Two adjacent edges in a geometric drawing of $K_6$ that cannot be extended to a plane Hamiltonian path. \ref{['fig:noprescribededges_b']} An illustration of \ref{['proposition:HP_pair_indep_edges']}: extending two independent edges in a geometric drawing of $K_n$ to a plane Hamiltonian path.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Conjecture 1: Rafla Rafla1988
  • Conjecture 2: Aichholzer, Orthaber, Vogtenhuber aov-2024-tcfhcsdcg
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Corollary 8
  • Theorem 9
  • Corollary 10
  • ...and 8 more