README.md

Circuit orderability of cographic matroids of complete and complete bipartite graphs

This paper classifies consistent reversible circuit orderings for two complete graph families.

• Paper (PDF)
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• BibTeX citation
• Version DOI: 10.5281/zenodo.22597680
• Concept DOI: 10.5281/zenodo.22597679
• Zenodo record
• Example checks
• Author: Carptopus (carptopus@163.com)
• Version: v0.1-beta, 7 September 2026
• Paper and documentation: CC BY 4.0; verification code: MIT

Main results

1. For every integer n >= 4, the cographic matroid M*(K_n) is orderable exactly when n = 4 or n = 6.
2. For all positive integers r, s, M*(K_{r,s}) is orderable exactly when min(r,s) <= 2.

The complete-graph proof reconstructs a closed simplicial surface from any consistent ordering, then uses a mod-two face equation and a link-degree obstruction. The bipartite classification adds a direct obstruction for K_{3,s} to previously known cases. These are structural proofs, not extrapolations from finite enumeration.

Relation to earlier work

The positive K6 example was already established in Projective-plane triangulations yield orderable cographic matroids. Triangle sums, peripheral cycles, and contractions for orderable cographic matroids supplies related construction and operation results. The present paper classifies arbitrary orderings on the two specified graph families; it does not replace those more general construction results or classify all cographic matroids.

Crenshaw–Oxley's four-connected regular theorem covers the cases r,s >= 4, and their K5 and K3,3 negative examples are explicitly credited. Their original paper and Crenshaw's 2023 dissertation are cited in the manuscript.

Verification and status

Run python verification/verify_examples.py with Python 3.10 or newer. No third-party package is required. The checker verifies all 7 bonds of the tetrahedral K4 and all 31 bonds of the given K6 complex, with destructive negative controls. It does not prove the infinite exclusion statements.

Internally checked preprint; external mathematical review and formal peer review are pending. OpenAI Codex assisted with research, verification, and writing; Carptopus is responsible for the manuscript.