HEXAGONbeta

Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Nikolay Ulyanov ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

The paper is an outcome of a continuous dialog between the author and AI (primarily, gpt, ocassionally claude as well). We were researching snark decomposition question, and wanted to understand, what if we specialize from bicritical snarks to more restrained hypohamiltonian or (conjecturally) permutation snarks. All results have been formalized in Lean 4 and could be found in author's github. Author has read the whole text, has checked the hypohamiltonian argument, and believes the permutation part is also correct, but needs some time to understand the proof properly.

References

  1. CS10
  2. GZ18
  3. MS19
  4. MS25
  5. BGHM13
  6. NS95

Version history

  1. v1Submitted by Nikolay UlyanovInitial depositCurrentOct 18, 2026