Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks
Every hypohamiltonian snark and every permutation snark has a unique multiset of cyclically 5-edge-connected factors of the same class under successive decompositions along cycle-separating 4-edge-cuts. We prove that hypohamiltonicity passes from a dot product to both of its snark factors. Together with the factorisation theorem for bicritical snarks, this answers a question of Goedgebeur and Zamfirescu. For permutation snarks, we extend the atom argument of Chladný and Škoviera using colourable cut multipoles and closure under their canonical completions.
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