Remarks on Barnette's Conjecture

TL;DR

This paper investigates the number of Hamilton cycles in a specific class of bipartite plane graphs related to Barnette's conjecture, providing a lower bound based on graph duality and face structure.

Contribution

It establishes a lower bound on the number of Hamilton cycles in cubic 3-connected bipartite plane graphs with certain face properties, advancing understanding of Barnette's conjecture.

Findings

Provides a lower bound of exponential size for Hamilton cycles.

Connects the number of Hamilton cycles to the dual graph's properties.

Offers insights into the structure of graphs related to Barnette's conjecture.

Abstract

Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{*}|}{\Delta^{2}{(P^{*})}}}$ different Hamilton cycles.