Hamiltonian cycles in planar cubic graphs with facial 2-factors, and a new partial solution of Barnette's Conjecture

TL;DR

This paper investigates Hamiltonian cycles in planar cubic graphs with facial 2-factors, transforming the problem into spanning trees of faces, and provides partial solutions to Barnette's Conjecture, especially for leapfrog extensions of certain bipartite graphs.

Contribution

It introduces a new approach linking Hamiltonicity to spanning trees of faces in contracted graphs and proves Hamiltonicity for leapfrog extensions of specific bipartite graphs, advancing Barnette's Conjecture.

Findings

Proves Hamiltonicity in leapfrog extensions of certain bipartite graphs.

Transforms Hamiltonian cycle problem into spanning tree of faces problem.

Provides partial solutions to Barnette's Conjecture.

Abstract

We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G/Q. In particular, we study the case where G is the leapfrog extension (called vertex envelope in (Discrete Math., 309(14):4793-4809, 2009)) of a plane cubic graph G_0. As a consequence we prove hamiltonicity in the leapfrog extension of planar cubic cyclically 4-edge-connected bipartite graphs. This and other results of this paper establish partial solutions of Barnette's Conjecture according to which every 3-connected cubic planar bipartite graph is hamiltonian. These results go considerably beyond Goodey's result on this topic (Israel J. Math., 22:52-56, 1975).