When I searching for interesting math problems. I find there is a graph theory conjecture called the Barnette's conjecture.
The statement is: Is every bipartite simple polyhedron Hamiltonian?
A early version which is proven to be false is: (Tait's conjecture) Is every 3-connected planar cubic graph has a Hamiltonian cycle through all its vertices?
The Tait's conjecture is disproved by Tutte graph.
But it is mentioned on the wiki page that there are smaller counterexamples. What are they? I couldn't find the paper.
And also, is there any other approaches approaching this problem?
Thank you very much for any form of help.