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Assume that we have a 3-connected cubic bipartite planar graph with a Hamiltonian cycle. That graph must have at least 4 Hamiltonian cycles, because of Theorem 1 and Theorem 10 of this paper.

I would like to know whether every edge of that graph is contained in at least one Hamiltonian cycle. It seems true for small 3-connected cubic bipartite planar graphs.

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According to Theorem 2.5 (on page 71) of Mathematical Combinatorics, Vol. 3/2014: international book series, what I'm asking for is true if and only if Barnette's conjecture holds true.

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