Published paper
Abstract
Selected proof-dependency source for the full recursive audit of arXiv:2605.16028v1.
Role in dependence graphs
Proof-critical source
Subgraphs versus Orientations: Infinite families of equidistributions
This paper is included only for the following marked statements:
- Theorem 3.2, decomposition step · arXiv:1408.3962v3, PDF p. 6Shows that the difference of cut-reversal-equivalent partial orientations decomposes as edge-disjoint directed cuts.
- Theorem 3.4, decomposition step · arXiv:1408.3962v3, PDF p. 7Gives the dual edge-disjoint directed-cycle decomposition.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Frozen exact arXiv:1408.3962v3
Spencer Backman. Partial Graph Orientations and the Tutte Polynomial. Advances in Applied Mathematics 94 (2018), 103–119.
The exact versioned arXiv PDF and source were frozen and inspected because the publisher landing page, not publisher full text, was acquired for this audit. This is an arXiv fallback and is not represented as the version of record.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The selected cut- and cycle-decomposition statements are correct and imply the Bernardi–Fang fact through the explicit two-successive-reversal specialization recorded below.
Cut decompositionCorrect · focal dependency verified
Theorem 3.2 supplies the cocycle half of the focal decomposition fact
arXiv:1408.3962v3, Theorem 3.2 proof, PDF p. 6; focal Fact 4.10, PDF p. 11.
The uniqueness argument proves that the symmetric difference of two partial orientations related by cut reversals decomposes into edge-disjoint directed cuts. For focal cuts and with no common arc, choose an orientation in which is directed and the -only edges already have their directions; reverse and then . Opposite-arc overlaps are reversed twice and cancel, while exactly the arcs in are reversed once. Backman's theorem then decomposes as required.
Official arXiv:1408.3962v3 ↗Cycle decompositionCorrect · focal dependency verified
Theorem 3.4 supplies the dual cycle half
arXiv:1408.3962v3, Theorem 3.4 proof, PDF p. 7; focal Fact 4.10, PDF p. 11.
The dual proof records that the symmetric difference of cycle-reversal-equivalent partial orientations decomposes as edge-disjoint directed cycles. The same explicit two-reversal construction with and makes their opposite-arc overlaps cancel and leaves exactly , so this theorem implies the cycle half of Fact 4.10.
Official arXiv:1408.3962v3 ↗02Proofs1 reported findingNo proof issues found
The selected route is internal: a Laplacian and chip-firing induction proves the cut statement, and the dual cycle-space argument gives the cycle statement.
Internal proof closureNo issue found in the reviewed proof
The cut decomposition closes without an earlier non-book theorem
arXiv:1408.3962v3, proof of Theorem 3.2, PDF p. 6.
Writing the indegree difference as a graph Laplacian vector, normalizing the firing vector, and reversing the directed cut on its support strictly reduces that support. The resulting induction yields disjoint directed cuts. No cited article is used in this selected proof step.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.