Barnette’s Conjecture Through the Lens of the $$Mod_{k}P$$ Complexity Classes

Abstract

In circa 2006, Feder & Subi established that Barnette’s 1969 conjecture, which postulates that all cubic bipartite polyhedral graphs are Hamiltonian, is true if and only if the Hamiltonian cycle decision problem for this class of graphs is polynomial time solvable (assuming \(\mathcal {P} \ne \mathcal {NP}\)). Here, we bridge the truth of Barnette’s conjecture with the hardness of a related set of decision problems belonging to the \(Mod_{k}P\) complexity classes (not known to contain \(\mathcal {NP}\)), where we are tasked with deciding if an integer k fails to evenly divide the Hamiltonian cycle count of a cubic bipartite polyhedral graph. In particular, we show that Barnette’s conjecture is true if there exists a polynomial time procedure for this decision problem when k can be any arbitrarily selected prime number. However, to illustrate the barriers for utilizing this result to prove Barnette’s conjecture, we also show that the aforementioned decision problem is \(Mod_{k}P\)-complete \(\forall k \in \left( 2 \mathbb {N}_{>0}+1\right) \), and more generally, that unless \(\mathcal {NP} = \mathcal {RP}\), no polynomial time algorithm can exist if k is not a power of two.