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Two classes of connectivity-related non-Hamiltonian 1-planar perfect graphs

Licheng Zhang, Shengxiang Lv, Yuanqiu Huang

Abstract

The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper, we investigate the non-Hamiltonicity of a broader class of graphs, specifically perfect graphs, under the constraint of 1-planarity, with a focus on connectivity of at most 5. We also propose some unsolved problems.

Two classes of connectivity-related non-Hamiltonian 1-planar perfect graphs

Abstract

The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper, we investigate the non-Hamiltonicity of a broader class of graphs, specifically perfect graphs, under the constraint of 1-planarity, with a focus on connectivity of at most 5. We also propose some unsolved problems.

Paper Structure

This paper contains 4 sections, 7 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Concluding Remarks
  4. Acknowledgment

Key Result

Theorem 1.1

Every $4$-connected $1$-planar chordal graph are Hamiltonian-connected.

Figures (2)

  • Figure 1: On the left is a quadrangulation $Q$, and on the right is the double-stellating quadrangulation $Q_s$.
  • Figure 2: A 1-planar drawing of $H_k$

Theorems & Definitions (16)

  • Theorem 1.1: Zhang
  • Theorem 1.2
  • Lemma 2.1: Hougardy
  • Lemma 2.2: page 287, West
  • Lemma 2.3: page 213, Bondy
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['main1']}
  • Claim 2.1
  • proof
  • ...and 6 more