ProofAtlas

Graph theory · cubic graphs · perfect matchings

Fan–Raspaud Conjecture

Collaboration beta

Can every finite bridgeless cubic graph admit three perfect matchings with no edge common to all three?

∀G(Gfinite bridgeless cubic)∃M0,M1,M2∈PM(G):M0∩M1∩M2=∅
Listed inDouglas West’s REGS page on consequences of the Berge–Fulkerson conjecture
Known results and sources
A low-text forest-green cover built around a cubic graph, with three perfect-matching layers distinguished by solid, doubled, and dashed line treatments and no edge carrying all three treatments.
The Fan–Raspaud Conjecture asks for three perfect matchings in every finite bridgeless cubic graph with no edge common to all three.

Research problem

Exact mathematical statement

∀G(Gfinite, bridgeless, and cubic)∃M0,M1,M2∈PM(G):M0∩M1∩M2=∅.\forall G\; (G \text{ finite, bridgeless, and cubic})\;\exists M_0,M_1,M_2\in\operatorname{PM}(G)\;: \; M_0\cap M_1\cap M_2=\varnothing.

the source develops exact-trace, quotient-packing, two-anchor local-lifting, and positive-slack routes, but explicitly leaves several gates open and claims no complete proof.

Problem infographic

Problem at a glance

Scientific infographic for the open Fan–Raspaud Conjecture on a deep forest-green background. An eight-vertex cube graph is drawn with three perfect matchings: gold connector edges M₀, emerald horizontal edges M₁, and cobalt vertical edges M₂. Short diagrams define cubic, bridgeless, and perfect matching. The central question asks whether every finite bridgeless cubic graph has three perfect matchings with M₀ ∩ M₁ ∩ M₂ = ∅. A comparison shows that an edge may belong to two matchings, while membership in all three is forbidden. The bottom states that the general case remains open.
The Fan–Raspaud conjecture asks whether every finite bridgeless cubic graph admits three perfect matchings with empty triple intersection. Pairwise overlap is allowed; the general case remains open.

Current mathematical picture

Where work on Fan–Raspaud Conjecture stands

Open conjecture

Revision 12 retains the exact Fan–Raspaud conjecture, applies its control-layer precedence over the Revision 11/10 archive, withdraws the parity-impossible prescribed odd-cut target, closes the two-contained c=9 lane conditionally on the inherited covariance and two shore audits, makes compaction class-relative, and leaves only labelled local routing kernels in that c=9 lane. No complete proof is claimed.

Strongest supported footholdDeep two-anchor lifting through six

The local theorem is packet-reported through six internal anchor edges, with computer-assisted cases awaiting replay and audit.

Evidence posture · Reported result
Leading routeExact-trace bi-anchor composition

Atomize an exact trace, pack two quotient matchings, and lift each prescribed pair through every atom using one of two retained anchors.

Route status · Active route
Useful failurec=9 fifth-excess synchronization

The Revision 10 route through a perfect matching disjoint from a prescribed odd-cut matching is parity-impossible and superseded by Revision 11's different lift.

Route status · Refuted route
Main reductionCommon-cut positive-slack clean core

Exact charge conservation, clean two-anchor atoms, and a bounded labelled port interface are retained under the exposed-common-cut hypothesis.

Evidence posture · Reported reduction
Completed special caseCorrect proper-union lift

For the proper-union geometry, form H=T[U]+xy. Two edge-disjoint perfect matchings of H lift to two edge-disjoint perfect matchings of T by replacing xy with ax,by when used and otherwise adding distinct copies of ab; no matching is synchronized to the false prescribed odd-cut target.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve bounded-interface absorption

Compose ordered two-matching boundary relations and anchor-rescue labels across arbitrarily long neutral corridors, yielding compatible transitions or strict descent.

Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Fan–Raspaud Conjecture in numbers

8.2kretained lines of mathematical investigation8,239 in the current working snapshot
Argument development
6,304 · 77%
Explored or eliminated routes
576 · 7%
Computational analysis
242 · 3%
Open obligations
525 · 6%
Definitions and setup
592 · 7%
29selected mapped statements14routes investigated9reported milestones10open questions9contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGenghua Fan, André RaspaudProposed1994Status checkedAug 2, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

28 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

28 selected steps

Scroll horizontally to explore the route

Working route overview for Fan–Raspaud ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.13-shore compatibility density — Depends on missing premise13-shore compatibilitydensity15-shore compatibility density — Depends on missing premise15-shore compatibilitydensityClass-relative finite-interface compaction — Depends on missing premiseClass-relativefinite-interface compactionFan–Raspaud conjecture — Depends on missing premiseFan–Raspaud conjectureAll-excess compressed-kernel theorem — ChallengedAll-excess compressed-kerneltheoremBounded-interface exceptional region — Depends on missing premiseBounded-interfaceexceptional regionc=9 selected-cut covariance threshold — Depends on missing premisec=9 selected-cut covariancethresholdCorrect proper-union lift — Depends on missing premiseCorrect proper-union liftCurrent provisional quotient frontier — Depends on missing premiseCurrent provisional quotientfrontierDistributional closure of the c=9 two-contained-cut lane — Depends on challenged claimDistributional closure ofthe c=9 two-contained-cutlaneExact-trace atomization — Depends on missing premiseExact-trace atomizationExhaustive-union compatibility gluing — Depends on challenged claimExhaustive-unioncompatibility gluingExact-trace bi-anchor composition — activeExact-trace bi-anchorcompositionRevision-12 quotient audit and c=3η−6 frontier — activeRevision-12 quotient auditand c=3η−6 frontierDeep two-anchor cyclic splitting — activeDeep two-anchor cyclicsplittingShallow localization and admissible recursion — activeShallow localization andadmissible recursionUniversal rooted-selector overlap using beta-only averaging — stoppedUniversal rooted-selectoroverlap using beta-onlyaveragingUnrestricted simultaneous two-port Kempe mobility — stoppedUnrestricted simultaneoustwo-port Kempe mobilityClose c=9 fifth excess using one shared bad edge — stoppedClose c=9 fifth excess usingone shared bad edgeUse one chosen deep anchor for every prescribed pair — stoppedUse one chosen deep anchorfor every prescribed pairClose exceptional fifth excess — OpenClose exceptional fifthexcessAudit c≥3η−5, then solve c=3η−6 kernels — OpenAudit c≥3η−5, then solvec=3η−6 kernelsExtend deep lifting beyond six anchors — OpenExtend deep lifting beyondsix anchorsProve prescribed-pair shallow localization — OpenProve prescribed-pairshallow localizationProve recursion admissibility and termination — OpenProve recursionadmissibility andterminationProve bounded-interface absorption — OpenProve bounded-interfaceabsorptionEnter the arbitrary positive-slack branch — OpenEnter the arbitrarypositive-slack branchReplay and independently audit load-bearing claims — OpenReplay and independentlyaudit load-bearing claims
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeExact-trace bi-anchor composition

Atomize an exact trace, pack two quotient matchings, and lift each prescribed pair through every atom using one of two retained anchors.

Route status · Active route
Active routeRevision-12 quotient audit and c=3η−6 frontier

Keep the Revision 10 piecewise envelope as the audit baseline, independently audit the source-reported provisional c≥3η−5 extension, then attack c=3η−6 and the separate c=7,9 fifth-excess exceptions; the old standalone c=3η−3 lane is superseded.

Route status · Active route
Active routeDeep two-anchor cyclic splitting

Extend the current work-reported theorem beyond six internal anchor edges while preserving protected roots and the second anchor.

Route status · Active route
Active routeShallow localization and admissible recursion

Use both anchors and injective routing forests, then certify that the smallest returned shore is an admissible strictly descending state.

Route status · Active route
Active routeCommon-cut positive-slack interface

Use exact conservation, clean atoms, port parity, and finite ordered boundary relations without assuming bounded internal order.

Route status · Active route
Active routeArbitrary positive-slack entry

Force exactness or a common cut, decrease slack, or expose a bounded non-common first-facet kernel before applying atomization.

Route status · Active route

Explored alternatives

Other routes

8 recorded
Refuted routec=9 fifth-excess synchronization

The Revision 10 route through a perfect matching disjoint from a prescribed odd-cut matching is parity-impossible and superseded by Revision 11's different lift.

Route status · Refuted route
Narrowed routeSuperseded quotient thresholds

The older c≥43, c≥15, c≥13, and c≥3η−1 thresholds remain separately checkable routes, but they are not current frontiers.

Route status · Narrowed route
Refuted routeUniversal fixed-anchor selector

Exact counterfamilies eliminate the universal one-anchor route; both anchors must remain in every local state.

Route status · Refuted route
Browse 5 more explored routes
Refuted routeUnrestricted two-port Kempe relocation

The unrestricted two-port claim is withdrawn; only source-status-aware one-port transfers remain.

Route status · Refuted route
Not yet justifiedBounded-order exceptional kernel

Bounded ports do not bound neutral-corridor order; the current conclusion is a bounded interface only.

Route status · Not yet justified
Useful but insufficientQuotient-only closure

Quotient matching selection cannot replace compatible local lifts, positive-slack entry, or global integration.

Route status · Useful but insufficient
Not yet justifiedAutomatic descent from a returned shore

A failure shore is paused at candidate-localizer status until factor-criticality, interfaces, and strict decrease are verified.

Route status · Not yet justified
Narrowed routec=9 fifth-excess after the parity repair

Treat the inherited covariance theorem and both shore arguments as explicit audit dependencies for the conditional two-contained-cut closure; the remaining mathematical work in this lane is the fully labelled local routing census.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementExact-trace atomization

For exact trace r=0, the state decomposes into 2κ connected odd factor-critical atoms whose contraction is a connected loopless c-regular odd-cut multigraph; global closure still requires compatible two-anchor local lifts.

Source-reported route statement · dependencies incomplete
Route statementPiecewise quotient-packing envelope

For the current work's connected loopless c-regular odd-cut quotient, two edge-disjoint perfect matchings are reported for η≤4 when c=7 or 9, η≤5 when c=11, and η≤⌊(c+2)/3⌋ for odd c≥13.

Source-reported route statement · dependencies incomplete
Route statementCommon-cut positive-slack conservation

After a common c-cut is exposed with 0<r<c, the current work reports exact charge conservation Σξᵢ+Γ=4(s−1)r and eight-unit quantization ξᵢ∈8ℤ≥0.

Source-reported route statement · dependencies incomplete
Route statementClean two-anchor atom core

Under the common-cut positive-slack hypotheses, the current work reports at least max{0,2κ−(s−1)r/2} clean factor-critical two-anchor c-atoms and at most (s²+s+2)r/2 labelled trace ports.

Source-reported route statement · dependencies incomplete
Route statementInjectively assigned routing forest

Every artificial-forest incidence must be assigned injectively to a distinct prescribed port at its vertex, with p(v)≥d_K(v)+1.

Source-reported route statement · dependencies incomplete
Route statementDeep two-anchor lifting through six

For a simple exact pole with disjoint internal anchors A,R, if A is deep and |A|≤6, every prescribed boundary pair is good relative to A or R; the five- and six-edge cases are computer-assisted.

Source-reported route statement · dependencies incomplete
Route statementTwo-anchor necessity

The exact counterfamilies 𝒪_c and 𝒳_c rule out a universal one-anchor theorem; in 𝒳_c a pair can fail for the first deep anchor and be rescued by the second.

Source-reported route statement · dependencies incomplete
Route statementAnnular prescribed-pair coverage

The annular transport matrix is a convex combination of injections, with every prescribed pair of distinct outer contacts appearing in a supported injection.

Source-reported route statement · dependencies incomplete
Route statementRouting failure returns a candidate localizer

Failure of an injectively assigned routing forest returns a proper near-tight shore, but it is only a candidate localizer until factor-critical re-atomization, interface preservation, and strict descent are proved.

Source-reported route statement · dependencies incomplete
Route statementBounded-interface exceptional region

For fixed s,r, the exceptional region communicates with the clean exact core through a bounded labelled port set and induces finitely many abstract single- and double-matching boundary relations; no bounded-order replacement follows yet.

Source-reported route statement · dependencies incomplete
Route statementGlobal bi-anchor composition

Two edge-disjoint quotient perfect matchings close an exact state only when their selected root pair is compatibly lifted in every atom relative to one of its two internal anchors.

Source-reported route statement · dependencies incomplete
Route statementCommon-cut atomization has limited scope

The common-cut positive-slack atomization does not cover arbitrary positive slack and cannot be invoked before a common cut has been exposed.

Source-reported route statement · dependencies incomplete
Route statementCurrent provisional quotient frontier

Revision 12 retains the piecewise Revision 10 envelope as the audit baseline, reports the stronger uniform bound c≥3η−5 as proved in-source but still awaiting independent checking, and makes c=3η−6 the next open quotient line; its first new general point is (c,η)=(15,7), while (9,5) remains separate.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

10 featured tasks
01
Prove bounded-interface absorption

Compose ordered two-matching boundary relations and anchor-rescue labels across arbitrarily long neutral corridors, yielding compatible transitions or strict descent.

Suggested move: Compute exact labelled relations and compose corridors without assuming bounded order; then construct a replacement, direct theorem, or strict recursive state.
Ready to work on
02
Close exceptional fifth excess

Audit the conditional two-contained-cut closure and complete the remaining labelled c=9 routing-kernel census; do not reuse the parity-impossible prescribed odd-cut target.

Suggested move: Independently audit the covariance and shore arguments, then enumerate the a=1 and a=2 routing kernels with every multiplicity, port, injection, odd-shore, and parent-profile field retained.
Ready to work on
03
Prove prescribed-pair shallow localization

For every prescribed outer pair, establish local goodness for one anchor or localize every compulsory common anchor edge behind a proper residual-tight shore.

Suggested move: Test both anchors, then run only injectively assigned routing forests and retain the smallest exact failure shore.
Ready to work on
04
Enter the arbitrary positive-slack branch

Without assuming a common c-cut, force an exact trace or common cut, strictly decrease slack, or expose a bounded non-common first-facet kernel.

Suggested move: Carry the exact canonical selection rule and analyze the first new facet before invoking atomization.
Ready to work on
05
Prove recursion admissibility and termination

Turn a localized shore into a smaller state while preserving factor-criticality, contacts, both anchors, and interfaces, and strictly decreasing the frozen potential.

Suggested move: Freeze the recursion potential and audit re-atomization on both sides of the smallest candidate shore.
Ready to work on
06
Extend deep lifting beyond six anchors

Prove a seven-plus-anchor two-anchor cyclic splitting theorem or retain exact rescued exceptional families.

Suggested move: Enumerate Eulerian seven-edge contraction types only after deep cut inequalities and protected-root reduction, preserving the second anchor.
Ready to work on
07
Audit c≥3η−5, then solve c=3η−6 kernels

Independently audit the Revision-12 c≥3η−5 quotient extension before using it, then solve the surviving four- and six-vertex kernels on c=3η−6 or return the first exact counterexample.

Suggested move: Replay the targeting and slot-allocation proof first; only after it passes, verify the generic, two-shore, and eight-vertex closures and enumerate the I+m≤3 six-vertex kernels.
Ready to work on
08
Replay and independently audit load-bearing claims

Reconstruct the post-Revision-8 quotient, routing, positive-slack, clean-core, and deep five-/six-anchor claims, returning the first exact failure if one appears.

Suggested move: Follow the proof-risk register in order and replay the archived five- and six-anchor programs before relying on their output.
Ready to work on
09
Classify the remaining c=9 routing kernels

Complete the labelled local routing-kernel census after the conditional two-contained-cut closure, retaining the profile and parent blocker certificate, labelled vertex set and order, every edge-copy multiplicity, all support/boundary ports, the artificial forest and injective incidence map, every odd-shore density inequality, blocker profile, colouring or smallest violating shore, and the lift back to the compressed quotient profile.

Suggested move: Enumerate and audit the a=1, |O|≤11 and a=2, |O|≤13 labelled kernels without collapsing multiplicities or interface labels.
Ready to work on
10
Integrate exact and positive-slack branches

Assemble the audited quotient, deep/shallow local, recursion, arbitrary-slack, and absorption results into an exact global argument.

Suggested move: Do not assemble until the applicable Q, D/S, P, and audit gates are closed.
Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

The conjecture remains unresolved for general bridgeless cubic graphs. Current work develops equivalent formulations and proves restricted graph classes; it does not supply a general proof.

[6][7][11]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedRecent work introduced regular colouring defect to quantify how close a triple of perfect matchings is to both empty intersection and full edge coverage.[7]
  2. PreprintMazzuoccolo and Zerafa gave an equivalent formulation via a variant of Petersen colourings and developed related restricted cases.[3]
  3. Peer reviewedThe conjecture was proved for bridgeless cubic graphs containing a 2-factor with at most two odd circuits.[5]
  4. PreprintFouquet and Vanherpe proved that a minimum counterexample would have at least 32 vertices.[1]
11 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFan–Raspaud conjecture
Logical consequenceBerge–Fulkerson conjecture

Any three perfect matchings from a six-matching Fulkerson cover have empty triple intersection, so Berge–Fulkerson implies Fan–Raspaud.

[2]
Equivalent formulationF4 Fano-colouring conjecture

The assertion that every bridgeless cubic graph admits a Fano colouring using at most four lines is equivalent to Fan–Raspaud.

[6]
Equivalent formulationthree complementary 2-factors cover all edges

Taking complements of the three perfect matchings gives an equivalent three-2-factor covering formulation.

[10]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Finite-interface compaction is bound to its intended classThe compaction theorem now requires a fixed gluing-compatible fragment class and every terminal-deletion relation needed by recursion, while retaining its non-effective limitation.

Changed the research frontierLater mathematical revision

Research stage 16
The c=9 two-contained-cut dependency chain is explicitThe successor binds the covariance threshold, corrected proper-union lift, two shore audits, exhaustive gluing, conditional distributional closure, and the fully labelled remaining routing census.

Changed the research frontierLater mathematical revision

Research stage 15
Odd-cut parity withdraws the prescribed c=9 targetRevision 11 proves the Revision 10 prescribed odd-cut disjoint-matching target false and invalidates every retained route, edge, derivation, milestone, cluster, and obligation that treated it as current.

Changed the research frontierLater mathematical revision

Research stage 14

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Claim connections clarified

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

13 mapped milestonesretained argument map

Browse all 13 mapped stages

  1. stage 1Exact-trace atomization and bi-anchor interface
  2. stage 2Historical fifth-excess threshold c≥43
  3. stage 3Historical fifth-excess threshold c≥15
  4. stage 4Historical fifth-excess threshold c≥13
  5. stage 5Specialized fifth excess through c≥11
  6. stage 6General quotient endpoint c≥3η−2
  7. stage 7One-port and routing-forest hypotheses repaired
  8. stage 8c=9 proper-union residual target isolated
  9. stage 9Common-cut positive-slack clean core
  10. stage 10Bounded interface replaces bounded-order kernel
  11. stage 11Deep two-anchor lifting through six
  12. stage 12Shallow route split into localization and admissible descent
  13. stage 13Current closing gates separated
Exact-trace atomization and bi-anchor interfaceThe current work organizes an exact trace into factor-critical atoms, a c-regular odd-cut quotient, and compatible two-anchor local lifts.

Mapped research milestoneInitial research sequence

Research stage 1
Historical fifth-excess threshold c≥43The archived stable-rank projection first reached fifth excess for c≥43.

Mapped research milestoneInitial research sequence

Research stage 2
Historical fifth-excess threshold c≥15Boundary pruning strengthened the archived fifth-excess threshold from c≥43 to c≥15.

Mapped research milestoneInitial research sequence

Research stage 3
Historical fifth-excess threshold c≥13A four-vertex equality repair strengthened the archived fifth-excess threshold to c≥13.

Mapped research milestoneInitial research sequence

Research stage 4
Specialized fifth excess through c≥11The specialized fifth-excess route now reaches odd c≥11, leaving c=7 and c=9 open.

Mapped research milestoneInitial research sequence

Research stage 5
General quotient endpoint c≥3η−2The defect–order route reaches c≥3η−2 and completes the general portion of the current piecewise quotient envelope.

Mapped research milestoneInitial research sequence

Research stage 6
One-port and routing-forest hypotheses repairedRevision 10 restricts boundary-defect preservation by source/target status and requires injective forest-incidence port assignments.

Mapped research milestoneInitial research sequence

Research stage 7
c=9 proper-union residual target isolatedThe proper-union branch reduces to a prescribed 13-cut matching problem in a 26-vertex 9-graph.

Mapped research milestoneInitial research sequence

Research stage 8
Common-cut positive-slack clean coreExact conservation and quantization leave many clean two-anchor atoms behind a bounded labelled trace-port interface.

Mapped research milestoneInitial research sequence

Research stage 9
Bounded interface replaces bounded-order kernelRevision 10 retains finite labelled boundary relations but withdraws the inference that the exceptional region has bounded order.

Mapped research milestoneInitial research sequence

Research stage 10
Deep two-anchor lifting through sixThe current work reports the two-anchor prescribed-pair theorem through six internal anchor edges and exact counterfamilies against one-anchor shortcuts.

Mapped research milestoneInitial research sequence

Research stage 11
Shallow route split into localization and admissible descentA valid routing failure now returns a candidate localizer, while re-atomization, interface preservation, and strict termination remain separate obligations.

Mapped research milestoneInitial research sequence

Research stage 12
Current closing gates separatedThe current frontier separates quotient exceptions, next kernels, deep and shallow lifting, recursion, absorption, arbitrary-slack entry, audit, and exact integration.

Mapped research milestoneInitial research sequence

Research stage 13

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

28 standing statements1 proposed statements9 mathematical milestones10 open questions2 narrowed routes17 conditional results5 completed special cases
Statements by mathematical role29 selected mapped statements
  • theorem candidate4 of 294
  • reduction11 of 2911
  • lemma10 of 2910
  • negative result3 of 293
  • definition1 of 291
Selected mathematical clusters7 mathematical clusters
Conjecture and exact-state architectureThe open conjecture, exact-trace atomization, and conditional global bi-anchor composition.7 displayed rows · 2 routes included
  • retained route statementFan–Raspaud conjecture
  • retained route statementExact-trace atomizationconditional
  • retained route statementGlobal bi-anchor compositionconditional
  • retained route statementQuotient packing alone is insufficientintermediate
  • DerivationThis is only the current work's conditional exact-branch assembly; uncovered quotient parameters, shallow lifting, positive-slack entry, and integration keep the conclusion unproved.proposed
  • Active routeExact-trace bi-anchor compositionAtomize an exact trace, pack two quotient matchings, and lift each prescribed pair through every atom using one of two retained anchors.
  • Useful but insufficientQuotient-only closureQuotient matching selection cannot replace compatible local lifts, positive-slack entry, or global integration.
Quotient packing and residual kernelsRetains the quotient envelope and explicit exceptional kernels, marks the prescribed odd-cut target refuted, and records the covariance/shore dependency chain for the conditional c=9 closure.21 displayed rows · 3 routes included
  • retained route statementPiecewise quotient-packing envelopeconditional
  • retained route statementAll-excess compressed-kernel theoremconditional
  • retained route statementSpecialized fifth-excess rangespecial case
  • retained route statementCurrent provisional quotient frontierconditional
  • ChallengeThe current work flags the endpoint Tutte inequality, surviving even-component edge, tight-star transfer, and two-shore colour availability as the first load-bearing points requiring independent reconstruction.unsupported step · open
  • Challengeindependent checking must check completeness of the four profiles, required blocker bundles, the one-shared-edge equality, and positivity of all six internal bundles.unsupported step · open
  • Research targetClose exceptional fifth excessopen
  • Research targetAudit c≥3η−5, then solve c=3η−6 kernelsopen
  • retained route statementOdd-cut parity refutes the prescribed 13-cut targetspecial case
  • retained route statementCorrect proper-union liftspecial case
  • retained route statementc=9 selected-cut covariance thresholdconditional
  • retained route statement13-shore compatibility densityspecial case
  • retained route statement15-shore compatibility densityspecial case
  • retained route statementExhaustive-union compatibility gluingconditional
  • retained route statementDistributional closure of the c=9 two-contained-cut laneconditional
  • DerivationThe selected-cut covariance forces positive pure-13 mass; every pure-13 state is proper-union or exhaustive-union, and the corrected lift plus compatibility-density gluing eliminate those geometries.active reported
  • ChallengeThe conditional closure remains provisional until the inherited covariance theorem, root-deletion matchability, both deficiency-one inequalities, the 13-shore fortieth-edge classification, the 15-shore six-row table, and the common-label gluing are independently audited.unsupported step · open
  • Research targetClassify the remaining c=9 routing kernelsopen
  • Active routeRevision-12 quotient audit and c=3η−6 frontierKeep the Revision 10 piecewise envelope as the audit baseline, independently audit the source-reported provisional c≥3η−5 extension, then attack c=3η−6 and the separate c=7,9 fifth-excess exceptions; the old standalone c=3η−3 lane is superseded.
  • Narrowed routeSuperseded quotient thresholdsThe older c≥43, c≥15, c≥13, and c≥3η−1 thresholds remain separately checkable routes, but they are not current frontiers.
  • Narrowed routec=9 fifth-excess after the parity repairTreat the inherited covariance theorem and both shore arguments as explicit audit dependencies for the conditional two-contained-cut closure; the remaining mathematical work in this lane is the fully labelled local routing census.
Deep and shallow two-anchor liftingThe deep theorem through six, the exact need for a second anchor, annular pair coverage, and the still-incomplete shallow descent interface.14 displayed rows · 4 routes included
  • retained route statementDeep two-anchor lifting through sixconditional
  • retained route statementOne fixed deep anchor sufficesconditional
  • retained route statementTwo-anchor necessityintermediate
  • retained route statementAnnular prescribed-pair coverageconditional
  • retained route statementRouting failure returns a candidate localizerconditional
  • ChallengeThe five- and six-anchor reductions depend on archived computation and still require replay plus an independent checking of the cyclic reduction and exceptional-family mapping.unsupported step · open
  • ComputationArchived deterministic programs for the five- and six-anchor two-anchor reductions, including the exceptional-family verifier.The current work reports the five- and six-anchor cases as computer-assisted; Revision 10 adds no certificate, and source report plus independent checking remain open. · reported unreproduced
  • Research targetExtend deep lifting beyond six anchorsopen
  • Research targetProve prescribed-pair shallow localizationopen
  • Research targetProve recursion admissibility and terminationopen
  • Active routeDeep two-anchor cyclic splittingExtend the current work-reported theorem beyond six internal anchor edges while preserving protected roots and the second anchor.
  • Active routeShallow localization and admissible recursionUse both anchors and injective routing forests, then certify that the smallest returned shore is an admissible strictly descending state.
  • Refuted routeUniversal fixed-anchor selectorExact counterfamilies eliminate the universal one-anchor route; both anchors must remain in every local state.
  • Not yet justifiedAutomatic descent from a returned shoreA failure shore is paused at candidate-localizer status until factor-criticality, interfaces, and strict decrease are verified.
Positive slack and bounded interfacesRetains positive-slack conservation and the bounded interface while adding the intended-class, gluing, terminal-deletion, and non-effectivity hypotheses needed for class-relative compaction.18 displayed rows · 3 routes included
  • retained route statementCommon-cut positive-slack conservationconditional
  • retained route statementClean two-anchor atom coreconditional
  • retained route statementExact-continent parityconditional
  • retained route statementFinite labelled boundary relationsconditional
  • retained route statementEdge-clean first-slack normal formsspecial case
  • retained route statementBounded ports imply bounded-order kernelconditional
  • retained route statementBounded-interface exceptional regionconditional
  • retained route statementCommon-cut atomization has limited scopeintermediate
  • ChallengeThe common-cut lift, copy-level charge nonnegativity, and treatment of even components and S-internal edges remain audit-sensitive.unsupported step · open
  • ChallengeThe current work calls for an independent check of the tight-cut classification, atom-count elimination, contamination injection, two-anchor existence, and port accounting.unsupported step · open
  • Research targetProve bounded-interface absorptionopen
  • Research targetEnter the arbitrary positive-slack branchopen
  • retained route statementComplete class-relative interface stateintermediate
  • retained route statementClass-relative finite-interface compactionconditional
  • Recorded relationshipFiniteness and context equivalence use the class-closed complete interface state, including the relations required for gluing and recursion.supports · reported by source
  • Active routeCommon-cut positive-slack interfaceUse exact conservation, clean atoms, port parity, and finite ordered boundary relations without assuming bounded internal order.
  • Active routeArbitrary positive-slack entryForce exactness or a common cut, decrease slack, or expose a bounded non-common first-facet kernel before applying atomization.
  • Not yet justifiedBounded-order exceptional kernelBounded ports do not bound neutral-corridor order; the current conclusion is a bounded interface only.
Corrections and superseded interfacesThe retained old and current claim versions for one-port transfer, routing-forest injectivity, and bounded-interface scope.11 displayed rows · 2 routes included
  • retained route statementOverstrong one-port boundary-count claimintermediate
  • retained route statementSource-status-aware one-port transferintermediate
  • retained route statementRouting forest without injective port assignmentintermediate
  • retained route statementInjectively assigned routing forestintermediate
  • retained route statementBounded ports imply bounded-order kernelconditional
  • retained route statementBounded-interface exceptional regionconditional
  • Useful failureUnrestricted simultaneous two-port Kempe mobilityreported failure
  • Useful failureInfer a bounded-order exceptional graph from bounded portsreported failure
  • ChallengeApplications must verify injective port assignment, odd-shore hypotheses, properness of a failure shore, and the parent-profile interpretation of routed colours.unsupported step · open
  • Refuted routeUnrestricted two-port Kempe relocationThe unrestricted two-port claim is withdrawn; only source-status-aware one-port transfers remain.
  • Not yet justifiedBounded-order exceptional kernelBounded ports do not bound neutral-corridor order; the current conclusion is a bounded interface only.
Eliminated and insufficient routesExact route failures prevent reuse of fixed-anchor, quotient-only, automatic-descent, arbitrary-scope, bounded-order, and premature-atomization shortcuts.17 displayed rows · 5 routes included
  • Useful failureUniversal rooted-selector overlap using beta-only averagingreported failure
  • Useful failureUnrestricted simultaneous two-port Kempe mobilityreported failure
  • Useful failureClose c=9 fifth excess using one shared bad edgereported failure
  • Useful failureUse one chosen deep anchor for every prescribed pairreported failure
  • Useful failureUniversal prescribed-pair theorem for admissible 11-vertex shoresreported failure
  • Useful failureTreat quotient packing as a complete proofreported failure
  • Useful failureTreat any local tight shore as recursive descentreported failure
  • Useful failureApply common-cut atomization to arbitrary positive slackreported failure
  • Useful failureInfer a bounded-order exceptional graph from bounded portsreported failure
  • Useful failureTreat a finite signature table as a replacement gadget theoremreported failure
  • Useful failureApply exact atomization before an exact or common-cut state is exposedreported failure
  • Useful failureDiscard the second internal anchorreported failure
  • Refuted routeUniversal fixed-anchor selectorExact counterfamilies eliminate the universal one-anchor route; both anchors must remain in every local state.
  • Refuted routeUnrestricted two-port Kempe relocationThe unrestricted two-port claim is withdrawn; only source-status-aware one-port transfers remain.
  • Not yet justifiedBounded-order exceptional kernelBounded ports do not bound neutral-corridor order; the current conclusion is a bounded interface only.
  • Useful but insufficientQuotient-only closureQuotient matching selection cannot replace compatible local lifts, positive-slack entry, or global integration.
  • Not yet justifiedAutomatic descent from a returned shoreA failure shore is paused at candidate-localizer status until factor-criticality, interfaces, and strict decrease are verified.
Open gates and independent checkingThe current remaining questions remain separate and no aggregate completion claim is inferred from their count.16 displayed rows · 7 routes included
  • Research targetClose exceptional fifth excessopen
  • Research targetAudit c≥3η−5, then solve c=3η−6 kernelsopen
  • Research targetExtend deep lifting beyond six anchorsopen
  • Research targetProve prescribed-pair shallow localizationopen
  • Research targetProve recursion admissibility and terminationopen
  • Research targetProve bounded-interface absorptionopen
  • Research targetEnter the arbitrary positive-slack branchopen
  • Research targetReplay and independently audit load-bearing claimsopen
  • Research targetIntegrate exact and positive-slack branchesblocked
  • Active routeExact-trace bi-anchor compositionAtomize an exact trace, pack two quotient matchings, and lift each prescribed pair through every atom using one of two retained anchors.
  • Active routeRevision-12 quotient audit and c=3η−6 frontierKeep the Revision 10 piecewise envelope as the audit baseline, independently audit the source-reported provisional c≥3η−5 extension, then attack c=3η−6 and the separate c=7,9 fifth-excess exceptions; the old standalone c=3η−3 lane is superseded.
  • Refuted routec=9 fifth-excess synchronizationThe Revision 10 route through a perfect matching disjoint from a prescribed odd-cut matching is parity-impossible and superseded by Revision 11's different lift.
  • Active routeDeep two-anchor cyclic splittingExtend the current work-reported theorem beyond six internal anchor edges while preserving protected roots and the second anchor.
  • Active routeShallow localization and admissible recursionUse both anchors and injective routing forests, then certify that the smallest returned shore is an admissible strictly descending state.
  • Active routeCommon-cut positive-slack interfaceUse exact conservation, clean atoms, port parity, and finite ordered boundary relations without assuming bounded internal order.
  • Active routeArbitrary positive-slack entryForce exactness or a common cut, decrease slack, or expose a bounded non-common first-facet kernel before applying atomization.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeCompose ordered two-matching boundary relations and anchor-rescue labels across arbitrarily long neutral corridors, yielding compatible transitions or strict descent.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • ordered two-matching signatures are preserved
  • anchor-rescue labels are preserved
  • the construction gives compatible lifts or strict descent

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointProve bounded-interface absorption

Fan–Raspaud Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Can every finite bridgeless cubic graph admit three perfect matchings with no edge common to all three?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references11 cited works · next context review by Nov 2, 2026

The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On Fan Raspaud Conjecturepreprint · accessed Aug 2, 2026
  2. 2
    On Fulkerson conjecturepreprint · accessed Aug 2, 2026
  3. 3
  4. 4
    Fulkerson’s Conjecture and Circuit Coversoriginal source · accessed Aug 2, 2026
  5. 5
    Sparsely intersecting perfect matchings in cubic graphspeer reviewed result · accessed Aug 2, 2026
  6. 6
  7. 7
  8. 8
  9. 9
    Consequences of the Berge-Fulkerson Conjectureauthoritative webpage · accessed Aug 2, 2026
  10. 10
    Snarksauthoritative webpage · accessed Aug 2, 2026
  11. 11
    2025 Developments in Combinatorics Workshopauthoritative webpage · accessed Aug 2, 2026

Important qualifications

  • Some papers alternate between simple bridgeless cubic graphs and bridgeless cubic multigraphs; public wording should match the exact ProofAtlas statement.
  • The scoped search did not verify a public proof-assistant formalization or a reusable exact-search dataset for the full conjecture.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image