Pancyclicity of almost-planar graphs
Santiago T. Adams, S. R. Kingan

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 vertices is pancyclic if it contains a cycle of every length from to , and it is Hamiltonian if it contains a cycle of length . A Hamiltonian path is a path of length 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 -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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Cellular Automata and Applications
