Published paper
Abstract
The paper defines bidirected orientations of signed graphs and proves that acyclic orientations correspond to regions of the signed-graphic hyperplane arrangement. Theorem 4.4 is the exact region-orientation input used by the focal paper.
Role in dependence graphs
Proof-critical source
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements
This paper is included only for the following marked statement:
- Theorem 4.4 · statement p. 371; vector proof setup pp. 366–371; alternative double-cover proof pp. 373–374Bijection between signed-graphic arrangement regions and acyclic signed orientations.
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Audit summary
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Exact reviewed source
Exact author-hosted journal reprint · European Journal of Combinatorics 12 (1991), 361–375
Thomas Zaslavsky. Orientation of signed graphs. European Journal of Combinatorics 12 (1991), 361–375.
Open audited source ↗01Statements3 reported findingsContains wrong statements
The orientation theorems and region correspondence are correct. The abstract contains one literal reversal: it says that regions correspond to cyclic orientations, while Theorem 4.4 and both proofs establish the acyclic-orientation correspondence. This is an unmistakable wording error rather than a theorem defect.
AbstractIncorrect wording · uniquely repairable
Cyclic must read acyclic
Journal page 361 · final sentence of the abstract
The abstract states that regions are in one-to-one correspondence with cyclic orientations. The paper's definition, Theorem 4.4, and independent Section 6 proof all say and prove acyclic orientations. Replacing cyclic by acyclic is the unique repair and leaves the body unchanged.
Exact author-hosted journal reprint ↗Theorem 4.4Correct
Signed-graphic regions correspond to acyclic orientations
Journal pages 370–371 · Theorem 4.4 and first proof
A point off the signed-graphic hyperplanes orients every link, half-edge, and negative loop by the signs of the appropriate coordinate functional. A directed signed circuit would force an impossible strict inequality cycle. Conversely the acyclic inequalities have a common solution, giving one arrangement region.
Theorems 3.1–5.1Correct
Switching, potential, and enumeration statements are consistent
Journal pages 364–372 · Sections 3–5
Switching changes both edge signs and local orientation coherently, directed circuits are switching invariant, and potential functions characterize acyclicity. Applying the region correspondence and Zaslavsky's arrangement count yields the stated chromatic evaluation without a missing loop or half-edge case.
02Proofs2 reported findingsCorrect
Both the oriented-matroid/vector proof and the independent double-cover proof of the central region theorem are correct. The remaining orientation lemmas cover all signed edge types and close without an unresolved case.
First proof of Theorem 4.4Correct
The strict signed inequalities characterize a nonempty region
Journal pages 366–371 · incidence-vector setup, Theorems 3.3 and 4.4
The incidence vectors represent the signed-graphic dependence matroid, and the oriented-matroid duality argument identifies acyclic sign choices with the chambers cut out by their hyperplanes. Theorem 4.4 then gives the required bijection between signed-graphic regions and acyclic orientations.
Section 6Correct
The double-cover proof independently verifies the same correspondence
Journal pages 373–374 · independent proof in Section 6
Section 6 realizes the signed-graphic arrangement as the cross-section of the ordinary graphical arrangement of the double covering graph. Proposition 6.1 shows that exactly lifted acyclic orientations meet and that their intersections are the signed regions, giving an alternative proof of Theorem 4.4.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.