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

Abstract

We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor . Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction ∕ G . In particular, we study the case where G is the leapfrog extension (called vertex envelope 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.

Matching Theory and Barnette's Conjecture

“…This conjecture has received considerable attention and is known to be true for some subclasses of B, but an approach to the problem in general continues to be elusive. See Goodey's [Goo75] paper from 1975 for the historically most significant example of a partial solution and see the work of Bagheri et al [BGFFS21] for a current partial solution which generalises Goodey's result. Significant effort has also been devoted to finding strengthenings of Conjecture 1.1; that is statements which also concern B but demand more than a simple Hamiltonian cycle.…”

Remarks on Barnette’s conjecture

“…Barnette, in 1969 ( [9], Problem 5), conjectured that every graph in P has a Hamilton cycle. In [5], Goodey proved that if a graph in P has only faces with 4 or 6 sides, then it is hamiltonian (see also Feder and Subi [4], and Bagheri, Feder, Fleischner and Subi [1]). Holton, Manvel and McKay [6] used computer search to confirm Barnette's conjecture for graphs up to 64 vertices.…”

Graphs with multi-$4$-cycles and the Barnette's conjecture

“…Goodey [5] proved that if a graph in B has only faces with 4 or 6 sides, then it is hamiltonian. Bagheri, Feder, Fleischner and Subi [2] study the existence of hamiltonian cycles in plane cubic graphs having a facial 2factor. The problem whether a cubic bipartite planar graph has a Hamilton cycle (without the assumption of 3-connectivity) is NP-complete, as shown by Takanori, Takao and Nobuji [11].…”