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Record W2074364524 · doi:10.1002/jgt.20571

On characterizing Vizing's edge colouring bound

2011· article· en· W2074364524 on OpenAlex
Penny Haxell, Jessica McDonald

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueJournal of Graph Theory · 2011
Typearticle
Languageen
FieldComputer Science
TopicAdvanced Graph Theory Research
Canadian institutionsSimon Fraser UniversityUniversity of Waterloo
Fundersnot available
KeywordsCombinatoricsMathematicsConjectureDomination analysisLogarithmBounded functionUpper and lower boundsMultiplicity (mathematics)GraphEdge coloringDiscrete mathematicsVertex (graph theory)Line graphGraph powerGeometry

Abstract

fetched live from OpenAlex

Abstract We consider multigraphs G for which equality holds in Vizing's classical edge colouring bound χ′( G )≤Δ + µ, where Δ denotes the maximum degree and µ denotes the maximum edge multiplicity of G . We show that if µ is bounded below by a logarithmic function of Δ, then G attains Vizing's bound if and only if there exists an odd subset S ⊆ V ( G ) with | S |≥3, such that | E [ S ]|>((| S | − 1)/2)(Δ + µ − 1). The famous Goldberg–Seymour conjecture states that this should hold for all µ≥2. We also prove a similar result concerning the edge colouring bound χ′( G )≤Δ + ⌈µ/⌊ g /2⌋⌉, due to Steffen (here g denotes the girth of the underlying graph). Finally we give a general approximation towards the Goldberg‐Seymour conjecture in terms of Δ and µ. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:160‐168, 2012

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.003
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.205
Threshold uncertainty score0.591

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0030.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0010.001
Science and technology studies0.0000.000
Scholarly communication0.0000.001
Open science0.0020.000
Research integrity0.0000.001
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.038
GPT teacher head0.279
Teacher spread0.241 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it