Minimum order of a planar hypohamiltonian graph

Determine whether the smallest planar hypohamiltonian graph has order 34, thereby proving or disproving the conjecture that no planar hypohamiltonian graph exists with fewer than 34 vertices.

Background

A planar hypohamiltonian graph is a non-Hamiltonian planar graph such that deleting any one vertex produces a Hamiltonian graph. The paper constructs planar hypohamiltonian graphs for every order at least 36 and discusses known examples of orders 34 and 37, making order 34 the smallest currently known order.

The authors support the conjecture using lower bounds, computational results, modular restrictions from graphs consisting of quadrilaterals and pentagons, and heuristic assumptions about the structure of any hypothetical smaller example. They identify several structural cases that would need to be excluded, but do not prove that every planar hypohamiltonian graph of order below 34 is impossible.