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arXiv · 2410.21239

Pancyclicity of almost-planar graphs

Santiago T. Adams·S. R. Kingan
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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.

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Santiago T. Adams, S. R. Kingan. 2024-10-28. Pancyclicity of almost-planar graphs. https://arxiv.org/abs/2410.21239

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