Published paper
Abstract
These lecture notes survey signed-graph theory with emphasis on the classical root systems and signed-graphic hyperplane arrangements. Lemma 4.3 identifies the signed chromatic polynomial with the arrangement characteristic polynomial.
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:
- Lemma 4.3 · arXiv:1303.2770v1 p. 21; author-formatted copy p. 20Equality of the signed chromatic polynomial and the signed-graphic hyperplane-arrangement characteristic polynomial.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact arXiv version 1 · explicit fallback from the 2012 journal article
Thomas Zaslavsky. Signed Graphs and Geometry. arXiv:1303.2770v1; published in Journal of Combinatorics, Information & System Sciences 37 (2012), 95–143.
The exact publisher PDF was not available. Exact arXiv version 1 and the author's journal-formatted copy were both checked; this report does not claim to audit an inaccessible publisher file.
Open audited source ↗01Statements3 reported findingsContains wrong statements
The survey's signed-graph, matroid, root-system, and hyperplane-arrangement theorems are correct. One displayed definition gives the negative-edge set with the positive sign fiber. The surrounding notation and every subsequent use determine the repair uniquely.
Negative-edge definitionIncorrect typo · uniquely repairable
The sign fiber has the wrong sign
arXiv:1303.2770v1 PDF page 4 · displayed definition of
The manuscript displays , which duplicates the positive-edge set. It must be . The immediately surrounding prose and every later formula use the corrected meaning, so this is a local notation error rather than a structural ambiguity.
Exact arXiv version 1 ↗Lemma 4.3Correct
The signed chromatic and arrangement characteristic polynomials agree
arXiv:1303.2770v1 PDF page 21 · Lemma 4.3 and preceding definitions; author version page 20
Both polynomials have the same spanning-subgraph expansion. For an edge set , the exponent is the number of balanced connected components, equal to the dimension of the intersection of the corresponding signed-graphic hyperplanes by Theorem 3.5 and Lemma 3.11. This is the exact focal source claim.
Sections 2–5Correct after the local sign repair
The signed-graphic geometry is consistent across all allowed edge types
arXiv:1303.2770v1 PDF pages 4–25 · signed graphs, vector models, polynomials, and arrangements
The survey explicitly allows links of both signs in parallel, as well as loops and half-edges where stated. Switching, balance, vector dependence, rank, and the arrangement equations all respect those conventions. This breadth is important: it is wider than the focal manuscript's simple signed-graph translation.
02Proofs3 reported findingsCorrect
The full survey proof chain was checked in the exact arXiv fallback and author copy. The rank, duality, subset-expansion, region, and coloring arguments are correct after the single displayed sign typo is repaired. No unresolved proof gap was found.
Theorem 3.5 and Lemma 3.11Correct
Vector rank and orthogonal-complement dimensions match balanced components
arXiv:1303.2770v1 PDF pages 16 and 19 · Theorem 3.5 and Lemma 3.11; author version pages 15 and 18
The signed incidence vectors have rank , where counts balanced connected components. Orthogonal complementation therefore gives an intersection flat of dimension . Theorem 3.5 explicitly points to the exact 1982 vector theorem, whose marked branch was separately checked and is unaffected by the known corrections.
Proof of Lemma 4.3Correct
The two subset expansions coincide term by term
arXiv:1303.2770v1 PDF page 21 · proof of Lemma 4.3; author version page 20
Inclusion–exclusion for proper signed colorings contributes . Whitney's formula for the signed-graphic arrangement contributes the same sign, while Lemma 3.11 identifies the same exponent as the flat dimension. Parallel edges are harmless because the signed graph and arrangement are treated with the stated multiplicity conventions.
Sections 4–5Correct
Region and root-system examples use the correct essential dimension
arXiv:1303.2770v1 PDF pages 19–25 · arrangement and root-system applications
Balanced components account for the lineality of nonessential arrangements, so Zaslavsky's region evaluation is applied in the ambient dimension with the correct polynomial factor. The type- and type- specializations then reproduce the standard Coxeter arrangements.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.