A note on optimal 2-planar graphs

TL;DR

This paper proves that all 4-connected optimal 2-planar graphs are Hamiltonian-connected and demonstrates that this property does not hold for 3-connected graphs, which can be non-Hamiltonian.

Contribution

It establishes the Hamiltonian-connectedness of 4-connected optimal 2-planar graphs and shows the sharpness of the connectivity condition with counterexamples.

Findings

All 4-connected optimal 2-planar graphs are Hamiltonian-connected.

There exist infinitely many 3-connected optimal 2-planar graphs that are non-Hamiltonian.

The 4-connectedness condition is necessary for Hamiltonian-connectedness in these graphs.

Abstract

In this note, we prove that every 4-connected optimal 2-planar graph is Hamiltonian-connected. Furthermore, we show that the 4-connectedness condition is sharp by constructing infinitely many 3-connected optimal 2-planar graphs that are non-Hamiltonian.