ProofAtlas

Graph theory · planar cubic graphs · Hamiltonian cycles

Barnette's Conjecture

Collaboration beta

Does every finite simple 3-connected planar graph that is both cubic and bipartite contain a Hamiltonian cycle?

Gfinite, simple, cubic, bipartite, planar, 3-connected⇒Gis Hamiltonian
Known results and sources
A planar cubic bipartite network surrounds a clean three-square ladder while an antique-gold route searches for a cycle through every visible vertex, evoking Barnette's conjecture without claiming a proof.
Barnette's Conjecture asks whether every finite simple cubic bipartite planar 3-connected graph has a Hamiltonian cycle.

Research problem

Exact mathematical statement

Gfinite, simple, cubic, bipartite, planar, and 3-connected⇒Ghas a Hamiltonian cycle.G\text{ finite, simple, cubic, bipartite, planar, and 3-connected} \quad\Longrightarrow\quad G\text{ has a Hamiltonian cycle}.

The retained revision-11 route proves several internal four-terminal and clean-ladder statements and records exact bounded computations, but it explicitly leaves the conjecture unresolved.

Problem infographic

Problem at a glance

Scientific explainer for the open Barnette Conjecture. A planar cube graph shows eight alternating circle and diamond vertices, degree three at every vertex, and a gold Hamiltonian cycle visiting all vertices once. Surrounding labels define cubic, bipartite, planar, and 3-connected, while asking whether every finite simple graph with these properties has such a cycle.
Barnette's Conjecture asks whether every finite simple cubic bipartite planar 3-connected graph has a Hamiltonian cycle. The cube graph provides a familiar example with such a cycle, but the general case remains open.

Current mathematical picture

Where work on Barnette's Conjecture stands

Open conjecture

Revision 40 reports a second-pass-audited global C4-expansion bridge, an all-order branch kernel, release of every biconnected first-failure state through b=8, and two specific b=9 pair-state releases. Barnette's conjecture remains unresolved: the remaining geometry-filtered b=9 orbit list, the b>=10 unbounded descent, tight-cut coordination, and a mark-preserving cycle merger are open. ProofAtlas has not executed or independently reproduced the current work attachments.

Strongest supported footholdAll-order clean-ladder signature

The current work proves the six parity-dependent boundary signatures and exact k+5 valid trace count for every clean ladder length.

Evidence posture · Reported result
Leading routeDirect terminal-exchange route

Seek a terminal matching whose selected dual is connected, using the side-bipartite plane exchange graph and exact exchange-face/factor-component correspondence.

Route status · Active route
Useful failureFixed-R_Q same-terminal rerouting

Cycle-run uniqueness rules out any nontrivial same-terminal one-path sibling with the same removed matching edges.

Route status · Eliminated route
Main reductionClean ladders compress within the Barnette class

Parity-matched square or domino replacement preserves graph class and Hamiltonicity, eliminating the three former audit exceptions as minimum-counterexample candidates.

Evidence posture · Reported reduction
Priority open bridgeProve the compression-critical two-site exit

For a minimum noncompressible simple-digon cap and one missing compression-critical state, prove an active lift, strict descent, exact boundary composition, corridor exit, or forbidden separator in the low-complexity hexagonal core.

Task status · Work already reported in progress
Latest mathematical updateRevision 40 records two audited local releases and the remaining exact frontier

The source reports an audited bridge and branch kernel, releases through b=8 plus two b=9 pair states, and leaves the full b=9 orbit list and b>=10 descent open.

Retained source record

Work mapped so far

Barnette's Conjecture in numbers

9.4kretained lines of mathematical investigation1,105 in the current working snapshot
Argument development
7,414 · 79%
Explored or eliminated routes
509 · 5%
Computational analysis
315 · 3%
Open obligations
601 · 6%
Definitions and setup
542 · 6%
39inventoried working statements19routes investigated13reported milestones17open questions12contribution-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 byDavid W. BarnetteProposed1969Status checkedAug 2, 2026
Source recordedAug 27, 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 Barnette's ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Barnette's conjecture remains open — Depends on missing premiseBarnette's conjectureremains openCorner-Stellation Reduction-Failure Lemma — Depends on missing premiseCorner-StellationReduction-Failure LemmaFirst unbounded branch-core frontier — Depends on missing premiseFirst unbounded branch-corefrontierMarked Leaf-Region Exit Lemma — Depends on missing premiseMarked Leaf-Region ExitLemmaNon-Ladder Terminal-Ear Complex Exit Lemma — Depends on missing premiseNon-Ladder Terminal-EarComplex Exit LemmaRemaining geometry-filtered b=9 orbit frontier — Depends on missing premiseRemaining geometry-filteredb=9 orbit frontierRooted cycle-merger and tight-cut coordination remain open — Depends on missing premiseRooted cycle-merger andtight-cut coordinationremain…Source-reported critical double-x2 b=9 overlap release — Depends on missing premiseSource-reported criticaldouble-x2 b=9 overlapreleaseSource-reported double outer-arm b=9 release — Depends on missing premiseSource-reported doubleouter-arm b=9 releaseSource-reported two-class hard-mark release through b=8 — Depends on missing premiseSource-reported two-classhard-mark release throughb=8Terminal Dual-Connectivity Lemma — Depends on missing premiseTerminal Dual-ConnectivityLemmaTwo-Defect Square-Orbit Exit Lemma — Depends on missing premiseTwo-Defect Square-Orbit ExitLemmaDirect terminal-exchange route — activeDirect terminal-exchangerouteClean-ladder compression — activeClean-ladder compressionSource-backed marked Tint route — activeSource-backed marked TintrouteTwo-defect square-orbit route — activeTwo-defect square-orbitrouteSame-terminal one-path rerouting with fixed removed set R_Q — stoppedSame-terminal one-pathrerouting with fixed removedset…Short-chord-only cap induction — stoppedShort-chord-only capinductionOrdinary-square-only induction — stoppedOrdinary-square-onlyinductionOne inductively admissible square or twin action per boundary state — stoppedOne inductively admissiblesquare or twin action perboundary…Prove the Non-Ladder Terminal-Ear Complex Exit Lemma — OpenProve the Non-LadderTerminal-Ear Complex ExitLemmaClose corner-stellation reduction failures — OpenClose corner-stellationreduction failuresClose longer lifted digons and even exchange faces — BlockedClose longer lifted digonsand even exchange facesHandle the synchronized two-path connected-unicycle case — BlockedHandle the synchronizedtwo-path connected-unicyclecaseProve Terminal Dual-Connectivity — BlockedProve TerminalDual-ConnectivityProve the Two-Defect Square-Orbit Exit Lemma — OpenProve the Two-DefectSquare-Orbit Exit LemmaProve the Marked Leaf-Region Exit Lemma — OpenProve the Marked Leaf-RegionExit LemmaRecover or reprove the arbitrary cap-replacement theorem — OpenRecover or reprove thearbitrary cap-replacementtheorem
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 routeDirect terminal-exchange route

Seek a terminal matching whose selected dual is connected, using the side-bipartite plane exchange graph and exact exchange-face/factor-component correspondence.

Route status · Active route
Active routeClean-ladder compression

Recognize maximal clean square ladders and replace them by the parity-equivalent square or domino before any non-ladder corridor analysis.

Route status · Active route
Active routeSource-backed marked Tint route

Use the Biedl–Kindermann Tint path, singleton bridge collapse, and marked balanced-tripod normal form as an independent route and cross-check every proposed matching against the direct exchange graph.

Route status · Active route
Active routeTwo-defect square-orbit route

At |S|=2, classify the first nonsquare interaction among three canonical channels between the two defect square complexes.

Route status · Active route
Active routeMarked leaf-region route

At |S| at least 4, use SDR marks to choose a canonical tripod and force an exit from the unique region-cycle hull.

Route status · Active route
Active routeCompression-critical hexagonal exit

Work only toward the five boundary states needed by compression, with fixed-exterior parity and exact square-site provenance controlling the low-complexity lollipop cases.

Route status · Active route
Active routeGeometry-filtered b=9 orbit and selector routeRoute status · Active route
Active routeUnbounded branch-core descent routeRoute status · Active route
Active routeTight-cut coordination and rooted cycle-merger routeRoute status · Active route

Explored alternatives

Other routes

10 recorded
Narrowed routeSimple four-edge lifted-digon branch

Reduce a simple lifted exchange digon to a Hamiltonian four-pole, exhaust active square and twin actions, compress clean ladders, and classify the remaining non-ladder square complexes.

Route status · Narrowed route
Route held in reserveLonger lifted digons and even exchange faces

Extend the one-path exchange-face exit beyond the simple four-edge lifted-digon case after the current non-ladder local theorem is closed.

Route status · Route held in reserve
Route held in reserveSynchronized two-path route

Handle the connected-unicycle selected-dual case only after the one-path exchange-face program is complete.

Route status · Route held in reserve
Browse 7 more explored routes
Eliminated routeFixed-R_Q same-terminal rerouting

Cycle-run uniqueness rules out any nontrivial same-terminal one-path sibling with the same removed matching edges.

Route status · Eliminated route
Eliminated routeShort-chord-only cap induction

The order-26 high-span diagnostic eliminates a proof strategy restricted to chord spans three or five.

Route status · Eliminated route
Eliminated routeSquare-only local induction

Order 12 eliminates ordinary-square reductions as a complete local action set; internal twins are indispensable.

Route status · Eliminated route
Eliminated routeOne local action per boundary state

Three retained encodings refute universal one-action liftability; ladder compression closes them as minimum candidates but does not erase the exact failures.

Route status · Eliminated route
Narrowed routeArbitrary terminal-ear synchronization

Revision 10's broad first target was narrowed after the former residual ears were identified as compressible clean ladders; only non-ladder complexes remain current.

Route status · Narrowed route
Not yet justifiedAutomatic full-graph C4/C5 transfer

The full-triangulation contraction-failure theorem cannot be applied automatically to a four-pole corner-stellation failure without an explicit hypothesis map.

Route status · Not yet justified
Not yet justifiedArchived arbitrary cap replacement

The older summary remains unusable because its exact profiles, gadgets, state sets, proof, and canonical artifacts are missing; revision 11 separately reproves the complete clean-ladder family.

Route status · Not yet justified

Route statements and reductions

Statements the next route can inspect and build on

Route statementTerminal Dual-Connectivity Lemma

For the common four-terminal exterior H, there is a terminal matching P whose selected dual H[P] is connected.

Source-reported route statement · dependencies incomplete
Route statementExchange graph is bipartite

Every terminal exchange graph Gamma_Q in the retained exterior inherits the two sides of the fixed Hamiltonian cycle, so it is bipartite and has no loops or odd cycles.

Source-reported route statement · dependencies incomplete
Route statementExchange faces equal factor components

Faces of Gamma_Q are canonically in bijection with components of H-P; in the two-terminal case bounded exchange faces correspond to cycle components.

Source-reported route statement · dependencies incomplete
Route statementClean ladder signature theorem

A clean k-square ladder has exactly six nonzero boundary signatures and k+5 valid local traces; odd k has the square signature and even k the domino signature.

Source-reported route statement · dependencies incomplete
Route statementBarnette-class clean-ladder compression

Replacing a clean facial ladder in a Barnette graph by one square for odd length or one domino for even length preserves the Barnette graph class and Hamiltonicity, and is strictly smaller for length at least three.

Source-reported route statement · dependencies incomplete
Route statementCorrected reported closure through order 22

The source reports that every one of the 9,163 valid encodings through order 22 is resolved by an inductively active square or twin action or by clean-ladder compression.

Source-reported route statement · dependencies incomplete
Route statementSource-backed exterior Tint path

When no two terminals are adjacent, adding BC in the kernel face yields an A-to-D Tint path containing BC and visiting every kernel-face boundary vertex.

Source-reported route statement · dependencies incomplete
Route statementMarked balanced-tripod normal form

Deleting BC from the retained Tint path yields two spanning paths outside an even balanced independent omitted set S of singleton tripods, each with a distinct marked interior attachment.

Source-reported route statement · dependencies incomplete
Route statementTransverse-region matching formula

Perfect matchings correspond to transverse cycle systems covering every degree-six quotient vertex, and the complementary-factor component count is one plus the nullity of the bipartite region graph.

Source-reported route statement · dependencies incomplete
Route statementTwo-defect square-orbit normal form

At Tint defect level |S|=2, the two defect square complexes are disjoint and connected, every pair in their orbit product is realizable, and three internally vertex-disjoint connecting paths exist.

Source-reported route statement · dependencies incomplete
Route statementTwo-Defect Square-Orbit Exit Lemma

The first nonsquare interaction among three canonical paths between the defect square complexes should yield a collision, strict region descent, a reducible two-factor state, a controlled cap or high-curvature site, or a forbidden separator.

Source-reported route statement · dependencies incomplete
Route statementMarked Leaf-Region Exit Lemma

For a minimum positive-defect marked Tint state, a transverse exchange at a minimum marked facial hull should destroy the unique region cycle, shrink the marked hull, annihilate adjacent centers, expose a controlled patch, or force a forbidden separator.

Source-reported route statement · dependencies incomplete
Route statementFixed-exterior matching extension

For a cubic graph split by four independent matching edges, every Hamiltonian cycle in the induced four-pole shore extends to a full perfect matching with the same exterior complementary-factor count.

Source-reported route statement · dependencies incomplete
Route statementCap projection of ambient square components

A facial square of the simple-digon cap belongs to an ambient isolated-square or domino component; a domino companion may lie across the cap boundary and then cannot be treated as an internal twin action.

Source-reported route statement · dependencies incomplete
Route statementRemaining geometry-filtered b=9 orbit frontier

The remaining b=9 task is to classify every geometrically realizable split-state pair orbit and then release each orbit by a direct lift, available selector slack, a joint selector, a closure-stable selector, or equality geometry.

Source-reported route statement · dependencies incomplete
Route statementFirst unbounded branch-core frontier

For b>=10 the known linear kernel exceeds the finite interfaces; a proof still needs a strict structural descent, expansion-pair absorption within the existing allocation, a nontrivial tight cut or end atom, or a pair-correct two-component spanning 2-factor.

Source-reported route statement · dependencies incomplete
Route statementRooted cycle-merger and tight-cut coordination remain open

A mark-preserving exchange must reduce component count or return a named separator, factorization, or two-cycle outcome, and marks lying in different tight-cut factors still require coordination.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

17 featured tasks
01
Build the V39-83 geometry-filtered b=9 orbit listSuggested move: Build the V39-83 geometry-filtered b=9 orbit list
Ready to work on
02
Prove the Marked Leaf-Region Exit Lemma

At marked Tint levels with at least four omitted vertices, eliminate or strictly shrink the unique marked region-cycle hull.

Suggested move: Use the injective bridge representatives to select the canonical first tripod at a minimum facial hull and audit each transverse exchange in the direct exchange graph.
Ready to work on
03
Prove the Two-Defect Square-Orbit Exit Lemma

Analyze the first nonsquare interaction among the three canonical connecting paths between the two defect square complexes.

Suggested move: Choose a canonical minimum region and record the Menger paths with endpoints and rotations before classifying the first nonsquare contact.
Ready to work on
04
Release any uncovered b=9 equality geometriesSuggested move: Release any uncovered b=9 equality geometries
Ready to work on
05
Prove V39-84 expansion-aware selector coverageSuggested move: Prove V39-84 expansion-aware selector coverage
Ready to work on
06
Coordinate tight-cut factors and a rooted two-cycle mergerSuggested move: Coordinate tight-cut factors and a rooted two-cycle merger
Ready to work on
07
Prove a strict branch-core descent for b>=10Suggested move: Prove a strict branch-core descent for b>=10
Ready to work on
08
Inventory cap squares by ambient component

Classify every cap square as internal isolated, internal domino, boundary-straddling domino, terminal-containing, or structurally failed before building an independent reduction family.

Suggested move: Record terminal incidence, ambient companion, common punctured base, replacement edges, and inductive graph checks for every candidate action.
Ready to work on
09
Close corner-stellation reduction failures

For every failed structurally admissible square or twin action, prove the separate Corner-Stellation Reduction-Failure Lemma or return an exact counterexample.

Suggested move: Inventory each failed action with corner rotation, distinguished traces, terminal cuts, and any full-graph C4/C5 hypothesis map.
Ready to work on
10
Constrain equality skeletons with curvature and exact safe substitutions

Use the residual face of length at least ten and at least eight facial squares to constrain the remaining equality skeletons, and prove every proposed substitution with exact boundary, safety/connectivity, marked-lift, and strict-descent data.

Suggested move: Derive the residual equality-skeleton restrictions and specify the first exact Barnette-safe substitution.
Ready to work on
11
Translate V39-78 through V40-85 into independently checkable formSuggested move: Translate V39-78 through V40-85 into independently checkable form
Ready to work on
12
Recover or reprove the arbitrary cap-replacement theorem

Materialize the exact profiles, gadgets, lifting proof, state sets, and canonical artifacts for the older arbitrary size-four cap replacement summary before using it.

Suggested move: Keep this auxiliary route separate from the self-contained clean-ladder theorem and reconstruct the missing profile table only if the direct frontier needs it.
Ready to work on
13
Prove the Non-Ladder Terminal-Ear Complex Exit Lemma

Classify each residual terminal-attached non-ladder square component after all active actions and ladder compressions and force one of the exact retained exits.

Suggested move: Build full-rotation square-adjacency components, record branches, cycles, repeated vertices and terminal incidences, and test direct lifts and exact terminal-cut compositions before corridor arguments.
Prerequisites still open
14
Prove Terminal Dual-Connectivity

Find a terminal matching of every common four-terminal exterior whose selected dual is connected.

Suggested move: Advance the simple-digon non-ladder exit while retaining the marked Tint program as an independent cross-check.
Blocked by the current route
15
Close longer lifted digons and even exchange faces

Extend the one-path exit beyond simple four-edge lifted digons to longer lifted digons and even facial exchange cycles of length at least four.

Suggested move: Finish the simple four-edge digon exit first, then construct the corresponding hull-exit mechanisms for longer lifted boundaries and longer even faces.
Blocked by the current route
16
Handle the synchronized two-path connected-unicycle case

After the one-path Exchange-Face Exit Lemma, treat the four-terminal matching case whose selected dual is connected unicyclic.

Suggested move: Defer until the full one-path exchange-face branch is closed, then use synchronized transition-surface data rather than one connected-shore matrix.
Blocked by the current route
17
Prove the compression-critical two-site exit

For a minimum noncompressible simple-digon cap and one missing compression-critical state, prove an active lift, strict descent, exact boundary composition, corridor exit, or forbidden separator in the low-complexity hexagonal core.

Suggested move: Audit the Grade-B channel normal forms, choose one missing critical state, and test the exact active/zero-sector alternatives of R13.5.
Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

It remains open whether every cubic, 3-connected, bipartite planar graph has a Hamiltonian cycle.

[7][8]
External progress

What the literature has established

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

  1. Peer reviewedApproximation work gives subhamiltonian structures for Barnette graphs while explicitly retaining the conjecture as open.[7]
  2. PreprintBrinkmann, Goedgebeur, and McKay verified the conjecture for all graphs up to at least 90 vertices.[4]
  3. PreprintAlt, Payne, Schmidt, and Wood proved new dual-coloring sufficient conditions and clarified Kelmans-style equivalent formulations.[3]
  4. Peer reviewedGoodey proved the conjecture for Barnette graphs whose face sizes are only four or six.[5]
12 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBarnette's conjecture
Weaker or relaxed formTait's conjecture

Barnette adds bipartiteness to Tait's false claim that every cubic 3-connected planar graph is Hamiltonian.

[6]
Equivalent formulationtree partition of simple even plane triangulations

In the planar dual, Barnette's conjecture is equivalent to partitioning the vertices of every simple even plane triangulation into two induced trees.

[1]
Solved special caseBarnette-Goodey bounded-face conjecture

The related claim for plane cubic 3-connected graphs with all face sizes at most six has been proved and does not settle Barnette's bipartite conjecture.

[6]
Equivalent formulationHamiltonicity of cubic planar braces

Matching-theoretic tight-cut decomposition yields an equivalent formulation for cubic planar braces.

[12]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • computation · source linked; not reproduced by ProofAtlasexhaustive graph verification

    A computer search verified every Barnette graph through 90 vertices as Hamiltonian.

    [4][7]

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.

Revision 40 records two audited local releases and the remaining exact frontierThe source reports an audited bridge and branch kernel, releases through b=8 plus two b=9 pair states, and leaves the full b=9 orbit list and b>=10 descent open.

Changed the research frontierLater mathematical revision

Revision-40 source ingested; not a claim of mathematical occurrence time
Cap compression is narrowed to the exact five-state frontierRevision 13 adds fixed-exterior parity and cap-square provenance while replacing two overstrong action assumptions by a scoped compression-critical exit problem.

Changed the research frontierLater mathematical revision

Barnette Conjecture revision 13 integrity and scope audit date ·

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

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.

12 mapped milestonesretained argument map

Browse all 12 mapped stages

  1. stage 1Short-kernel minimum-counterexample architecture
  2. stage 2Tint source recovered and marked tripod form established
  3. stage 3One-path exchange geometry sharpened
  4. stage 4Simple lifted digons reduced to Hamiltonian four-poles
  5. stage 5Four-pole frontier reported through order 26
  6. stage 6Exact square-and-twin action calculus
  7. stage 7One-action lifting refuted by three retained encodings
  8. stage 8Arbitrary terminal-ear synchronization staged
  9. stage 9Revision 11 corrects activity, faciality, and exception geometry
  10. stage 10All-order clean-ladder signatures established
  11. stage 11Clean ladders compress and close the former exceptions
  12. stage 12Current frontier refined to non-ladder square complexes
Short-kernel minimum-counterexample architectureThe retained route passes from a minimum cubic planar brace to square or domino short-factor states and a common four-terminal exterior.

Mapped research milestoneInitial research sequence

Research stage 1
Tint source recovered and marked tripod form establishedA mapped literature Tint path and singleton bridge collapse produce an even balanced set of marked tripods as an independent route.

Mapped research milestoneInitial research sequence

Research stage 2
One-path exchange geometry sharpenedThe exchange graph is bipartite, fixed-deletion siblings are rigid, and its faces are exactly the complementary-factor components.

Mapped research milestoneInitial research sequence

Research stage 3
Simple lifted digons reduced to Hamiltonian four-polesA simple lifted exchange digon cuts off a balanced Hamiltonian four-pole with exact terminal and corner-stellation structure.

Mapped research milestoneInitial research sequence

Research stage 4
Four-pole frontier reported through order 26The retained computation reports no nonuniversal simple-digon four-pole through 26 vertices and retains an order-26 witness against short-chord-only induction.

Mapped research milestoneInitial research sequence

Research stage 5
Exact square-and-twin action calculusInternal-square curvature, active/zero-sector bijections, and twin-or-ear certificates provide the local action framework.

Mapped research milestoneInitial research sequence

Research stage 6
One-action lifting refuted by three retained encodingsThe order-10 reflections and order-18 synchronized witness refute the claim that one tested square or twin action activates every state.

Mapped research milestoneInitial research sequence

Research stage 7
Arbitrary terminal-ear synchronization stagedRevision 10 provisionally made synchronized terminal-ear exit the first local theorem after the one-action failures.

Mapped research milestoneInitial research sequence

Research stage 8
Revision 11 corrects activity, faciality, and exception geometryThe current work separates three activity notions, proves cap C4 faciality, narrows automatic C4/C5 transfer, and independently reconstructs the three exceptions.

Mapped research milestoneInitial research sequence

Research stage 9
All-order clean-ladder signatures establishedEvery clean k-square ladder has six parity-dependent nonzero boundary signatures and exactly k+5 valid traces.

Mapped research milestoneInitial research sequence

Research stage 10
Clean ladders compress and close the former exceptionsParity-matched square or domino replacement preserves the Barnette class and Hamiltonicity and eliminates all three former one-action exceptions as minimum candidates.

Mapped research milestoneInitial research sequence

Research stage 11
Current frontier refined to non-ladder square complexesThe first local obligation now begins only after active actions and clean-ladder compression and targets branching, cyclic, self-contacting, reduction-blocked, and terminal-cut complexes.

Mapped research milestoneInitial research sequence

Research stage 12

Detailed research inventory

Claims, milestones, and routes in the current map

This inventory covers all currently cataloged mathematical statements in the research notes.

30 standing statements9 proposed statements13 mathematical milestones17 open questions2 narrowed routes13 conditional results
Statements by mathematical role39 mapped statements
  • reduction8 of 398
  • theorem candidate12 of 3912
  • lemma9 of 399
  • negative result2 of 392
  • equivalence5 of 395
  • computational claim3 of 393
Complete mathematical inventory10 mathematical clusters
Conjecture and minimum-counterexample architectureThe open conjecture, conditional short-kernel reduction, common four-terminal exterior, and global selected-dual target.7 displayed rows · 1 route included
  • retained route statementBarnette's conjecture remains open
  • retained route statementSquare-or-domino short-kernel reductionconditional
  • retained route statementTerminal Dual-Connectivity Lemmaconditional
  • Recorded relationshipThe retained minimum-counterexample program seeks to lift Hamiltonicity from the generated square or domino short-factor states.supports · reported by source
  • Recorded relationshipA connected selected dual produces the required Hamiltonian path or two-path complement in the common exterior and closes the short-kernel lift.supports · reported by source
  • Research targetProve Terminal Dual-Connectivityblocked
  • Active routeDirect terminal-exchange routeSeek a terminal matching whose selected dual is connected, using the side-bipartite plane exchange graph and exact exchange-face/factor-component correspondence.
One-path exchange geometrySide-tree bipartition, fixed-deletion rigidity, factor-component faces, and the remaining longer-face obligations.11 displayed rows · 4 routes included
  • retained route statementExchange graph is bipartiteintermediate
  • retained route statementFixed-deletion one-path rigidityintermediate
  • retained route statementExchange faces equal factor componentsintermediate
  • Recorded relationshipThe side bipartition, reroute rigidity, and exact face interpretation narrow the one-path terminal-dual problem to lifted digons and longer even faces.supports · reported by source
  • Useful failureSame-terminal one-path rerouting with fixed removed set R_Qreported failure
  • Research targetClose longer lifted digons and even exchange facesblocked
  • Research targetHandle the synchronized two-path connected-unicycle caseblocked
  • Active routeDirect terminal-exchange routeSeek a terminal matching whose selected dual is connected, using the side-bipartite plane exchange graph and exact exchange-face/factor-component correspondence.
  • Eliminated routeFixed-R_Q same-terminal reroutingCycle-run uniqueness rules out any nontrivial same-terminal one-path sibling with the same removed matching edges.
  • Route held in reserveLonger lifted digons and even exchange facesExtend the one-path exchange-face exit beyond the simple four-edge lifted-digon case after the current non-ladder local theorem is closed.
  • Route held in reserveSynchronized two-path routeHandle the connected-unicycle selected-dual case only after the one-path exchange-face program is complete.
Simple-digon Hamiltonian four-polesCap normal form, terminal cuts, parity, universal compression, and the reported order-26 frontier.11 displayed rows · 2 routes included
  • retained route statementSimple lifted-digon four-pole normal formconditional
  • retained route statementTerminal localization of 2-cutsintermediate
  • retained route statementFive-state four-pole parityintermediate
  • retained route statementUniversal-cap compressionconditional
  • retained route statementReported Hamiltonian four-pole frontier through order 26computational
  • DerivationBipartiteness restricts the first bounded exchange obstruction to a digon or longer even face; the exact face/component correspondence turns a simple lifted digon into one Hamiltonian factor component with four boundary edges.active reported
  • DerivationThe known AC state and five-state parity identify when all replacement-gadget states occur; replacing the cap then gives a smaller Barnette graph whose Hamiltonian trace lifts back through K.active reported
  • ComputationSource-reported exhaustive Hamiltonian-cycle/chord enumeration of simple-digon Hamiltonian four-poles through order 26.The source reports 12,279,722,829 labeled encodings, 154,054 valid cap encodings, and no nonuniversal cap through 26 vertices; a separate implementation reproduces counts through order 18. · reported unreproduced
  • Useful failureShort-chord-only cap inductionreported failure
  • Narrowed routeSimple four-edge lifted-digon branchReduce a simple lifted exchange digon to a Hamiltonian four-pole, exhaust active square and twin actions, compress clean ladders, and classify the remaining non-ladder square complexes.
  • Eliminated routeShort-chord-only cap inductionThe order-26 high-span diagnostic eliminates a proof strategy restricted to chord spans three or five.
Square, twin, and active-lift calculusCurvature action sites, exact active/zero decomposition, twin or ear certificates, bounded active audit, and the retained one-action failures.15 displayed rows · 2 routes included
  • retained route statementInternal-square curvature supplyintermediate
  • retained route statementSquare active/zero-sector decompositionintermediate
  • retained route statementBlocked square edge gives a twin or terminal earintermediate
  • retained route statementReported square-and-twin active audit through order 22computational
  • retained route statementOne action activates every boundary state
  • Recorded relationshipCurvature supplies square action sites, the active decomposition specifies liftability, and blocked edges identify twin or terminal-ear alternatives tested by the audit.supports · reported by source
  • Recorded relationshipThe two reflected order-10 encodings and the order-18 AD|BC witness remain exact failures of the one-action statement.refutes · reported by source
  • DerivationThe exact analyzer inventories facial square and twin sites, filters to inductively admissible reduced four-poles, and counts only covers that use a replacement edge for each boundary state.active reported
  • ChallengeTwo reflected order-10 encodings and one order-18 encoding are boundary-universal but have boundary states not activated by any single tested ordinary-square or internal-twin action.counterexample · reported resolved
  • Useful failureOrdinary-square-only inductionreported failure
  • Useful failureOne inductively admissible square or twin action per boundary statereported failure
  • ComputationSource-reported active ordinary-square and internal-twin reduction audit for every valid encoding of orders 10 through 22.All 9,163 encodings have a valid square or twin action; three labeled encodings fail to activate all six states by one inductively admissible action. · reported unreproduced
  • ComputationIndependent source-reported NetworkX/Python reconstruction of the two order-10 and one order-18 former active-audit exceptions.The audit records all six state counts, inductive-active vectors, facial squares, terminal 2-cuts, and an exact AD|BC composition at the order-18 cut {7,16}. · reported unreproduced
  • Eliminated routeSquare-only local inductionOrder 12 eliminates ordinary-square reductions as a complete local action set; internal twins are indispensable.
  • Eliminated routeOne local action per boundary stateThree retained encodings refute universal one-action liftability; ladder compression closes them as minimum candidates but does not erase the exact failures.
Clean-ladder signatures and compressionCap faciality, all-order ladder signatures, parity-equivalent gadgets, Barnette-class replacement, and corrected bounded closure.12 displayed rows · 1 route included
  • retained route statementEvery cap 4-cycle is facialintermediate
  • retained route statementClean ladder signature theoremintermediate
  • retained route statementParity-dependent ladder boundary equivalenceintermediate
  • retained route statementBarnette-class clean-ladder compressionconditional
  • retained route statementCorrected reported closure through order 22computational
  • Recorded relationshipCap faciality identifies the audited C4 chains as clean facial ladders, while the signature theorem determines their exact boundary behavior.supports · reported by source
  • Recorded relationshipMatching the six boundary pairings permits trace substitution in both directions; the separate graph-class argument preserves the Barnette hypotheses.supports · reported by source
  • Recorded relationshipThe three one-action exceptions each contain clean three-square ladders and are eliminated as minimum-counterexample candidates by compression.supports · reported by source
  • DerivationThe all-order signature identifies the square or domino replacement by parity; matching port colors, relative deletion profiles, and trace substitution preserve bipartiteness, 3-connectivity, and Hamiltonicity.active reported
  • DerivationThe two order-10 exceptions contain one clean L3 and the order-18 exception contains two disjoint clean L3 patches, so ladder compression handles exactly the cases missed by the one-action audit.active reported
  • ComputationDeterministic local checker for the clean-ladder boundary signature and trace-count formula for lengths 1 through 16.The source reports agreement with the six-signature parity theorem and the k+5 trace count for every checked length 1 through 16. · reported unreproduced
  • Active routeClean-ladder compressionRecognize maximal clean square ladders and replace them by the parity-equivalent square or domino before any non-ladder corridor analysis.
Current non-ladder simple-digon frontierThe superseded broad terminal-ear target, the current non-ladder exit, corner-failure scope, terminal-cut composition, and the exact next obligations.12 displayed rows · 3 routes included
  • retained route statementTerminal-Ear Synchronization Lemmaconditional
  • retained route statementNon-Ladder Terminal-Ear Complex Exit Lemmaconditional
  • retained route statementCorner-Stellation Reduction-Failure Lemmaconditional
  • ChallengeRevision 11 shows that the three former residual ear configurations are clean ladders and replaces the arbitrary terminal-ear target with a non-ladder square-complex exit lemma.overclaimed scope · reported resolved
  • Useful failureAutomatic transfer of the full-triangulation C4/C5 contraction certificatereported failure
  • Useful failureTreating neutral square or hexagon migration as induction descentreported failure
  • Research targetProve the Non-Ladder Terminal-Ear Complex Exit Lemmaopen
  • Research targetClose corner-stellation reduction failuresopen
  • DerivationOnce active actions and clean ladders are removed, the source classifies the remaining square components by branching, cycles, self-contact, terminal attachment, or reduction failure and assigns exact exit obligations to each class.proposed
  • Narrowed routeSimple four-edge lifted-digon branchReduce a simple lifted exchange digon to a Hamiltonian four-pole, exhaust active square and twin actions, compress clean ladders, and classify the remaining non-ladder square complexes.
  • Narrowed routeArbitrary terminal-ear synchronizationRevision 10's broad first target was narrowed after the former residual ears were identified as compressible clean ladders; only non-ladder complexes remain current.
  • Not yet justifiedAutomatic full-graph C4/C5 transferThe full-triangulation contraction-failure theorem cannot be applied automatically to a four-pole corner-stellation failure without an explicit hypothesis map.
Independent marked Tint programSource-backed Tint paths, marked tripods, transverse region cycles, the two-defect square orbit, and two current exit lemmas.12 displayed rows · 3 routes included
  • retained route statementSource-backed exterior Tint pathconditional
  • retained route statementMarked balanced-tripod normal formconditional
  • retained route statementTransverse-region matching formulaconditional
  • retained route statementTwo-defect square-orbit normal formconditional
  • retained route statementTwo-Defect Square-Orbit Exit Lemmaconditional
  • retained route statementMarked Leaf-Region Exit Lemmaconditional
  • DerivationThe Tint path and singleton bridge collapse produce marked tripods; contracting the dimer and tripod factor then yields the transverse quotient and the unique region-cycle obstruction.active reported
  • Research targetProve the Two-Defect Square-Orbit Exit Lemmaopen
  • Research targetProve the Marked Leaf-Region Exit Lemmaopen
  • Active routeSource-backed marked Tint routeUse the Biedl–Kindermann Tint path, singleton bridge collapse, and marked balanced-tripod normal form as an independent route and cross-check every proposed matching against the direct exchange graph.
  • Active routeTwo-defect square-orbit routeAt |S|=2, classify the first nonsquare interaction among three canonical channels between the two defect square complexes.
  • Active routeMarked leaf-region routeAt |S| at least 4, use SDR marks to choose a canonical tripod and force an exit from the unique region-cycle hull.
Archived gaps and retained route failuresThe arbitrary cap-replacement archive gap and the exact negative witnesses that later refinements must preserve rather than erase.15 displayed rows · 6 routes included
  • retained route statementArchived arbitrary cap theorem is not invokableintermediate
  • Useful failureInvoking the archived arbitrary cap-replacement summary as a black boxreported failure
  • Useful failureSame-terminal one-path rerouting with fixed removed set R_Qreported failure
  • Useful failureShort-chord-only cap inductionreported failure
  • Useful failureOrdinary-square-only inductionreported failure
  • Useful failureOne inductively admissible square or twin action per boundary statereported failure
  • Useful failureAutomatic transfer of the full-triangulation C4/C5 contraction certificatereported failure
  • Useful failureTreating neutral square or hexagon migration as induction descentreported failure
  • Research targetRecover or reprove the arbitrary cap-replacement theoremopen
  • Not yet justifiedArchived arbitrary cap replacementThe older summary remains unusable because its exact profiles, gadgets, state sets, proof, and canonical artifacts are missing; revision 11 separately reproves the complete clean-ladder family.
  • Eliminated routeFixed-R_Q same-terminal reroutingCycle-run uniqueness rules out any nontrivial same-terminal one-path sibling with the same removed matching edges.
  • Eliminated routeShort-chord-only cap inductionThe order-26 high-span diagnostic eliminates a proof strategy restricted to chord spans three or five.
  • Eliminated routeSquare-only local inductionOrder 12 eliminates ordinary-square reductions as a complete local action set; internal twins are indispensable.
  • Eliminated routeOne local action per boundary stateThree retained encodings refute universal one-action liftability; ladder compression closes them as minimum candidates but does not erase the exact failures.
  • Not yet justifiedAutomatic full-graph C4/C5 transferThe full-triangulation contraction-failure theorem cannot be applied automatically to a four-pole corner-stellation failure without an explicit hypothesis map.
Revision-13 compression frontierFixed-exterior parity, ambient square provenance, two corrected overstrong routes, and the compression-critical hexagonal exit target.7 displayed rows · 1 route included
  • retained route statementFixed-exterior matching extensionintermediate
  • retained route statementCap projection of ambient square componentsintermediate
  • Useful failureRequire all six nonzero domino-order boundary states before cap compressionreported failure
  • Useful failureApply the multi-square zero-set product identity to any visually disjoint square sitesreported failure
  • Research targetProve the compression-critical two-site exitin progress reported
  • Research targetInventory cap squares by ambient componentopen
  • Active routeCompression-critical hexagonal exitWork only toward the five boundary states needed by compression, with fixed-exterior parity and exact square-site provenance controlling the low-complexity lollipop cases.
Revision-40 recorded research mapThe exact open target, source-reported bridge and audited local releases, remaining finite and unbounded frontiers, explicit no-revisit registry, and prioritized work queue.26 displayed rows · 3 routes included
  • retained route statementBarnette's conjecture remains open
  • retained route statementSource-reported global C4 expansion-pair bridge
  • retained route statementSource-reported audited branch-kernel architecture
  • retained route statementSource-reported two-class hard-mark release through b=8
  • retained route statementSource-reported critical double-x2 b=9 overlap release
  • retained route statementSource-reported double outer-arm b=9 release
  • retained route statementRemaining geometry-filtered b=9 orbit frontier
  • retained route statementFirst unbounded branch-core frontier
  • retained route statementRooted cycle-merger and tight-cut coordination remain open
  • Recorded relationshipThe revision-40 source states this dependency explicitly in its controlling architecture or current frontier.supports · reported by source
  • Recorded relationshipThe revision-40 source states this dependency explicitly in its controlling architecture or current frontier.supports · reported by source
  • Recorded relationshipThe revision-40 source states this dependency explicitly in its controlling architecture or current frontier.supports · reported by source
  • Recorded relationshipThe revision-40 source states this dependency explicitly in its controlling architecture or current frontier.supports · reported by source
  • Recorded relationshipThe revision-40 source states this dependency explicitly in its controlling architecture or current frontier.supports · reported by source
  • Research targetBuild the V39-83 geometry-filtered b=9 orbit listopen
  • Research targetProve V39-84 expansion-aware selector coverageopen
  • Research targetRelease any uncovered b=9 equality geometriesopen
  • Research targetProve a strict branch-core descent for b>=10open
  • Research targetCoordinate tight-cut factors and a rooted two-cycle mergeropen
  • Research targetConstrain equality skeletons with curvature and exact safe substitutionsopen
  • Research targetTranslate V39-78 through V40-85 into independently checkable formopen
  • Useful failureRevision-40 no-revisit registryreported failure
  • ComputationThe revision-40 source reports a finite critical-overlap closure check and an inherited SHA-256 continuity comparison.The source reports ten normalized one/two-closure records with zero failures and matching hashes for all 28 controlling V38/V39 certificate files; ProofAtlas retained the named artifacts inert and did not reproduce either check. · reported unreproduced
  • Active routeGeometry-filtered b=9 orbit and selector routeGeometry-filtered b=9 orbit and selector route
  • Active routeUnbounded branch-core descent routeUnbounded branch-core descent route
  • Active routeTight-cut coordination and rooted cycle-merger routeTight-cut coordination and rooted cycle-merger route
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 bridgeFor a minimum noncompressible simple-digon cap and one missing compression-critical state, prove an active lift, strict descent, exact boundary composition, corridor exit, or forbidden separator in the low-complexity hexagonal core.

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.

  • Every listed low-complexity alternative is closed or reduced by a declared minimality tuple.
  • The resulting transition realizes the chosen state and uses the required replacement edge.

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Does every finite simple 3-connected planar graph that is both cubic and bipartite contain a Hamiltonian cycle?

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Sources and references12 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 Barnette's Conjecture and H^{+-} propertypreprint · accessed Aug 2, 2026
  2. 2
    Barnette's Conjecturepreprint · accessed Aug 2, 2026
  3. 3
    Thoughts on Barnette's Conjecturepreprint · accessed Aug 2, 2026
  4. 4
    The Minimality of the Georges-Kelmans Graphpreprint · accessed Aug 2, 2026
  5. 5
    Remarks on Barnette's conjectureoriginal source · accessed Aug 2, 2026
  6. 6
  7. 7
    Approximating Barnette's Conjectureauthoritative webpage · accessed Aug 2, 2026
  8. 8
    Barnette's conjectureencyclopedia · accessed Aug 2, 2026
  9. 9
    List of unsolved problems in mathematicsencyclopedia · accessed Aug 2, 2026
  10. 10
    Formal Conjectures repositoryformalization · accessed Aug 2, 2026
  11. 11
    Barnette's Conjecture — MathWorldoriginal source · accessed Aug 2, 2026
  12. 12
    Matching theory and Barnette's conjectureauthoritative webpage · accessed Aug 2, 2026

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  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

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