Hamiltonian cycles in some family of cubic $3$-connected plane graphs

TL;DR

This paper introduces a construction method for Hamiltonian cycles in a specific class of cubic, 3-connected plane graphs with face size constraints, addressing Barnette's conjecture in particular cases.

Contribution

It provides a novel construction technique for Hamiltonian cycles in certain cubic 3-connected plane graphs with face size limitations.

Findings

Constructs Hamiltonian cycles in graphs with specified face size conditions.

Addresses special cases related to Barnette's conjecture.

Offers a method applicable to graphs with a face g meeting certain criteria.

Abstract

Barnette conjectured that all cubic $3$-connected plane graphs with maximum face size at most $6$ are hamiltonian. We provide a method of construction of a hamiltonian cycle (in dual terms) in an arbitrary cubic, $3$-connected plane graph possessing such a face $g$ that every face incident with $g$ has at most $5$ edges and every other face has at most $6$ edges.