bibkey: harary1953notion
authors: Frank Harary
year: 1953
title: On the notion of balance of a signed graph
doi: 10.1307/mmj/1028989917
claim: Classical signed-graph balance formulation acknowledged for the binary cycle criterion, not the entire finite-graph theorem bundle.
strata_touched:
  - D5/S3/Fourier/CharacterSelection/SimpleGraphCycleSpace
license: citation-only
triage: anchor

Signed balance and the binary cycle criterion

RRO67.1 cites this article for the classical signed-graph balance formulation. For a binary edge label y, assign sign +1 to zero and -1 to one. Vanishing parity on every cycle means that every cycle has positive sign product. A vertex lift y({u,v}) = x(u) + x(v) modulo two expresses the edge sign as the product of the endpoint signs. This is the classical core acknowledged by the formal theorem’s literature reference.

The full theorem also gives arbitrary-anchor fiber bijections, constructions for every supplied spanning forest, cardinalities, and a fair-product probability. These clauses and the Lean proof are not attributed as a bundle to this paper. RRO67.1 reuses only the binary graph argument from RRO61.5; the full 2-adic valuation classification is outside this result. No external theorem number or copied proof is used here.

Locator

• DOI: https://doi.org/10.1307/mmj/1028989917
• Michigan Mathematical Journal 2(2) (1953), 143-146, as cited by RRO67.1.
• Crossref confirms author, title, year, journal, volume, issue, and DOI. This metadata verification does not verify the paper’s mathematical text.
• The primary article has not been independently read: the publisher’s article and PDF endpoints did not yield readable article text in the available retrievals. Attribution of the classical core follows the explicit source citation; exact article theorem scope is unverified.

This is a citation and original summary, not a source copy or a claim of permission to redistribute the article.