On the algorithmic complexity of finding hamiltonian cycles in special classes of planar cubic graphs

TL;DR

This paper investigates the computational complexity of finding Hamiltonian cycles in specific classes of planar cubic graphs, linking it to Barnette's Conjecture and exploring algorithmic challenges in these graph classes.

Contribution

It analyzes the complexity of Hamiltonian cycle problems in planar cubic graphs with facial 2-factors, connecting the problem's difficulty to Barnette's Conjecture and proposing new insights.

Findings

Hamiltonicity in certain planar cubic graphs is NP-complete if Barnette's Conjecture is false.

The paper establishes a link between facial 2-factors and Hamiltonian cycle existence.

Complexity results depend on the truth of Barnette's Conjecture.

Abstract

It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite graphs is an NP-complete problem.