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The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture

From Theorem 1.1 of the Hadwiger paper: deleting at most three vertices from its graphs gives graphs with $\alpha=2$ on $4t-1$ vertices whose connected-matching number stays below $t$ for all large $t$ , so the sharp conjecture fails. The weaker conjecture of the same authors, that $\mathrm{cm}(G)\ge c_0|V(G)|$ for some absolute $c_0>0$ , is not settled: the bound $\mathrm{cm}<m/100$ leaves smaller coefficients open, and the paper says so.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture · Furedi-Gyarfas-Simonyi connected matching

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

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Named evidence

VibeMathed record: The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the historical section of the Hadwiger counterexample paper, which derives this disproof from its Theorem 1.1 in a short remark, read against the conjecture as cited (Furedi-Gyarfas-Simonyi 2005). Not refereed. The 7 October 2026 update added the Comparator challenge HadwigerCounterexample; its exists_counterexamples statement (read here) gives arbitrarily large $m$ -vertex graphs with $\alpha(G)\le2$ and connected-matching number below $m/100$ . It does not state the conjecture's form (graphs on $4t-1$ vertices with $\alpha=2$ , matching size $t$ ); the short deletion step is not formalized, so the entry stays unreviewed. The disproof applies only for sufficiently large t.

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The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Furedi-Gyarfas-Simonyi 2005, CPC 14

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Lean statement (supporting): HadwigerCounterexample - Theorem 1.1 only, not the 4t-1 form

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean statement (supporting): HadwigerCounterexample - Theorem 1.1 only, not the 4t-1 form

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Lean statement (supporting): HadwigerCounterexample - Theorem 1.1 only, not the 4t-1 form evidence for this event

Furedi-Gyarfas-Simonyi 2005, CPC 14 evidence for this event

The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture parent of this event

The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture evidence for this event

VibeMathed record: The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture evidence for this event

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