Every prism over a 3-connected planar graph is hamiltonian.

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite simple graph $G$, its prism $G\square K_2$ has vertices $(v,i)$ with $v\in V(G)$ and $i\in\{0,1\}$. Its edges join $(u,i)$ to $(v,i)$ when $uv\in E(G)$ and join $(v,0)$ to $(v,1)$. A graph is $3$-connected if it has at least four vertices and remains connected after deletion of fewer than three vertices. A Hamilton cycle visits every vertex once.

Statement recorded in the supplied source.

The recorded conjecture asks whether $G\square K_2$ has a Hamilton cycle for every $3$-connected planar graph $G$. [S2]

Short English statement

Is the prism of every 3-connected planar graph Hamiltonian?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3176 (OPG-47285). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Every prism over a 3-connected planar graph is hamiltonian., original node 47285, statement and mathematical discussion. https://www.openproblemgarden.org/op/every_prism_over_a_3_connected_planar_graph_is_hamiltonian.