bibkey: ghanbarisamal2026facial
authors: Babak Ghanbari, Robert Šámal
year: 2026
title: "Facial diagrams and cycle double cover"
doi: 10.48550/arXiv.2605.01410
url: https://arxiv.org/abs/2605.01410v1
claim: "Twisting a regular edge makes it good singular; the facial-diagram analysis describes the reverse change and preservation of the twisted bad singular edge."
strata_touched:
  - D5/S3/Combinatorics/PerfectMatchings/FiniteBandSwitch
license: citation-only
triage: anchor

Facial edge types and the twist operation

Verified locator

Babak Ghanbari and Robert Šámal, Facial diagrams and cycle double cover, arXiv:2605.01410v1, submitted 2 May 2026; EUROCOMB’25 extended abstract. DOI: 10.48550/arXiv.2605.01410. Versioned record: https://arxiv.org/abs/2605.01410v1. Primary text: https://arxiv.org/html/2605.01410v1, Section 2 Proposition 2.1, Section 3 Lemma 3.1, and Section 3.1 item 8. Those locators support the twist properties quoted below; the probability conclusion additionally requires a twist-invariant law.

Source statement

Primary source: https://arxiv.org/html/2605.01410v1. This is the EUROCOMB’25 extended abstract submitted to arXiv on 2 May 2026.

Section 1 defines the three edge types using facial walks and traversal direction. Section 2, Proposition 2.1 describes merging the two facial walks incident to a regular edge. Section 3, Lemma 3.1 describes twisting singular links and their crossing links. Section 3.1, item 8 states that a twisted regular link is good singular.

Attribution and scope

The ribbon-band operation and its edge-type changes are established mathematics. A general permutation-orbit formalization supplies reusable formal infrastructure, not a claim of external novelty. Its orbit effect must be derived from the actual changed matching, rather than assumed as classifier equivariance.

Equal good and regular probabilities additionally require a sampling law invariant under the single-edge twist. The source’s Section 4 exact-thirds conjecture leaves its probability law unspecified. Twist symmetry alone does not imply that either expectation is one third of the edge count.