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

TL;DR

This paper explores the non-Hamiltonian properties of certain 1-planar perfect graphs with connectivity up to five, extending previous work on Hamiltonian cycles in higher-connected 1-planar graphs.

Contribution

It introduces new results on non-Hamiltonicity in 1-planar perfect graphs with low connectivity and proposes open problems for further research.

Findings

Identifies classes of 1-planar perfect graphs that are non-Hamiltonian

Extends understanding of Hamiltonian properties in low-connectivity 1-planar graphs

Proposes open problems for future investigation

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.