Pancyclicity of almost-planar graphs

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.

Figures (11)

• Figure 1: The Möbius ladder $V_{2k}$, where $k\ge 3$.
• Figure 2: The bicycle wheel $B_n$, for $n\ge 5$.
• Figure 3: The operation of attaching a fan in a triangle
• Figure 4: Sticking fans in $K_{3,3}"'$
• Figure 5: Cycles of size $k+3$ and $k+5$ in $V_{2k}$, where $k$ is even

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4