# A counterexample to prism-hamiltonicity of 3-connected planar graphs

## TL;DR

This paper constructs a specific counterexample disproving the conjecture that all 3-connected planar graphs are prism-hamiltonian, challenging a long-standing assumption in graph theory.

## Contribution

The paper provides the first known counterexample to the conjecture that every 3-connected planar graph is prism-hamiltonian.

## Key findings

- Counterexample to the conjecture is constructed
- Disproves the universal property of prism-hamiltonicity in 3-connected planar graphs
- Challenges previous assumptions in graph theory

## Abstract

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. A graph $G$ is hamiltonian if there exists a spanning cycle in $G$, and $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. In [M.~Rosenfeld, D.~Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973), 389--394] the authors conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.