Pancyclicity of almost-planar graphs

TL;DR

This paper investigates the pancyclic properties of almost-planar graphs, establishing conditions under which these graphs contain cycles of all lengths and are Hamiltonian-connected, based on their connectivity levels.

Contribution

It proves that 3-connected almost-planar graphs are pancyclic if they contain a triangle, and 4-connected almost-planar graphs are both pancyclic and Hamiltonian-connected.

Findings

3-connected almost-planar graphs are pancyclic iff they contain a triangle

4-connected almost-planar graphs are both pancyclic and Hamiltonian-connected

Characterization of pancyclicity in almost-planar graphs based on connectivity

Abstract

A non-planar graph is almost-planar if either deleting or contracting any edge makes it planar. A graph with $n$ vertices is pancyclic if it contains a cycle of every length from $3$ to $n$, and it is Hamiltonian if it contains a cycle of length $n$. A Hamiltonian path is a path of length $n$ and a graph with a Hamiltonian path between every pair of vertices is called Hamiltonian-connected. In 1990, Gubser characterized the class of almost-planar graphs. This paper explores the pancyclicity of these graphs. We prove that a $3$-connected almost-planar graph is pancyclic if and only if it has a cycle of length 3. Furthermore, we prove that a 4-connected almost-planar graph is both pancyclic and Hamiltonian-connected.