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Journal of Applied Mathematics and Computation

DOI：http://dx.doi.org/10.26855/jamc.2023.06.012

# A Note on Barnette’s Conjecture

Zijun Xiao

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China.

*Corresponding author: Zijun Xiao

Date: July 31,2023　Hits: 336

### Abstract

In 1969, Barnette conjectured that every 3-connected cubic planar bipartite graph is Hamiltonian. We obtain two results to help under-stand Barnette’s conjecture. The first result is inspired by the generation theorem of 3-connected cubic planar bipartite graphs, which is the work of Holton, Manvel and McKay. We define two operations called VO and EOR and prove that all graphs which are generated by the two operations from a unique graph of order 8 are Hamiltonian. We deduce also an equivalent description for the 3-connectivity of simple cubic plane bipartite graphs by the recent research hotspots of quasi spanning tree of faces. We prove that every simple cubic plane bipartite graph G is 3-connected if and only if the contraction graph HR(G) is 2-connected and 3-edge-connected, which meaning that if every 2-connected, 3-edge-connected planar graph of whose all vertex degrees are even and more than four has one quasi spanning tree of faces, then every 3-connected cubic planar bipartite graph is Hamiltonian.

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### Full-Text HTML

A Note on Barnette’s Conjecture

How to cite this paper: Zijun Xiao. (2023) A Note on Barnette’s Conjecture. Journal of Applied Mathematics and Computation7(2), 304-311.

DOI: http://dx.doi.org/10.26855/jamc.2023.06.012