A sufficient condition for cubic 3-connected plane bipartite graphs to be hamiltonian

TL;DR

This paper provides new sufficient conditions for cubic 3-connected plane bipartite graphs to be Hamiltonian, extending previous results and establishing exponential lower bounds on the number of Hamilton cycles under certain face-structure conditions.

Contribution

It proves that no face in such graphs having more than four big neighbors guarantees Hamiltonicity, and shows exponential growth in Hamilton cycles when each vertex is incident with both small and big faces.

Findings

Graphs with no face having more than four big neighbors are Hamiltonian.

Under certain face-incident conditions, the number of Hamilton cycles grows exponentially.

Extends partial results related to Barnette's conjecture for specific face configurations.

Abstract

Barnette's conjecture asserts that every cubic $3$-connected plane bipartite graph is hamiltonian. Although, in general, the problem is still open, some partial results are known. In particular, let us call a face of a plane graph big (small) if it has at least six edges (it has four edges, respectively). Goodey proved for a $3$-connected bipartite cubic plane graph $P$, that if all big faces in $P$ have exactly six edges, then $P$ is hamiltonian. In this paper we prove that the same is true under the condition that no face in $P$ has more than four big neighbours. We also prove, that if each vertex in $P$ is incident both with a small and a big face, then~$P$ has at least $2^{k}$ different Hamilton cycles, where $k = \left\lceil\frac{|B|-2}{4\Delta(B) - 7}\right\rceil$, $|B|$ is the number of big faces in $P$ and $\Delta(B)$ is the maximum size of faces in $P$. 15 pages