ProofAtlas

Planar graph drawing · rational distances · arithmetic geometry

Harborth's Conjecture

Collaboration beta

Many local gadgets and graph classes are controlled; the unresolved step is fitting them together while preserving planarity and target neighborhoods.

‖pu-pv‖∈ℤ(uv∈E)
Known results and sources
Problem-first open-research thumbnail for Harborth's Conjecture, showing an unresolved mathematical structure without a completion mark or navigation arrow.
Can every planar graph be drawn with integer edge lengths? The page presents source-reported partial structure without claiming a proof.

Research problem

Exact mathematical statement

Every finite planar graph has a crossing-free straight-line drawing in which every edge has integer length.

‖pu-pv‖∈ℤ(uv∈E)\lVert p_u-p_v\rVert\in\mathbb Z\quad(uv\in E)

This remains open; the source reports partial results and route constraints, not a complete proof.

Problem infographic

Problem at a glance

Three-panel source-bound explainer for Harborth's Conjecture: exact target, strongest retained result, and the unresolved closing bridge.
The source separates the exact target, retained partial results, and the still-open bridge.

Current mathematical picture

Where work on Harborth's Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureNaive shelling incidence graph

Shelling variables need not remain locally incident after pivot substitution. Relative owner routing, boundary absorption, flip transversality, and degree-at-most-five induction remain viable.

Route status · Narrowed route
Main reductionCurrent reduction

Triangulate, allocate faces or use an unspecialized shelling, and recursively close coupled blocks while retaining private or laminar arithmetic reserve.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve sparse factorization before shelling substitution.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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 We removed a duplicate or outdated task or route step.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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Harborth's Conjecture in numbers

1.3kretained lines of mathematical investigation1,316 in the current working snapshot
Argument development
1,099 · 84%
Explored or eliminated routes
29 · 2%
Computational analysis
23 · 2%
Open obligations
88 · 7%
Definitions and setup
77 · 6%
7selected mapped statements1routes investigated3open questions3contribution-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.

Status checkedAug 21, 2026
Source recordedAug 21, 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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Harborth'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.Can every planar graph be drawn with integer edge lengths? — Depends on missing premiseCan every planar graph bedrawn with integer edgelengths?Current reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetGaussian product engine — Depends on missing premiseGaussian product engineRelative private reserve — Depends on missing premiseRelative private reserveRepaired Hall surplus — Depends on missing premiseRepaired Hall surplusSubstitution-spread obstruction — Depends on missing premiseSubstitution-spreadobstructionNaive shelling incidence graph — stoppedNaive shelling incidencegraphProve sparse factorization before shelling substitution. — OpenProve sparse factorizationbefore shellingsubstitution.Absorb exported boundary corner factors recursively. — OpenAbsorb exported boundarycorner factors recursively.Close pentagon and degenerate insertion cases. — OpenClose pentagon anddegenerate insertion cases.
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.

Explored alternatives

Other routes

1 recorded
Narrowed routeNaive shelling incidence graph

Shelling variables need not remain locally incident after pivot substitution. Relative owner routing, boundary absorption, flip transversality, and degree-at-most-five induction remain viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove sparse factorization before shelling substitution.Suggested move: Derive a bounded-width factor graph.
Ready to work on
02
Absorb exported boundary corner factors recursively.Suggested move: Prove the two-boundary absorption lemma.
Ready to work on
03
Close pentagon and degenerate insertion cases.Suggested move: Use the degree-at-most-five route as an independent hedge.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 21, 2026
Current statusOpen conjecture

Current literature proves substantial special cases, including recent quadrilateral progress, but the general integer-length drawing conjecture remains open.

[1][2]
External progress

What the literature has established

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

  1. PreprintOn the Harborth Conjecture. Part I supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[1]
  2. Peer reviewedHarborth's Conjecture for 4-Regular Planar Graphs supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[2]
2 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHarborth's Conjecture
Related problemHarborth's Conjecture

Current literature proves substantial special cases, including recent quadrilateral progress, but the general integer-length drawing conjecture remains open.

[1][2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal statement matching the exact public target and all quantifiers.
  • Formalization targetFormal libraries for the principal mathematical structures used by the strongest route.
  • Formalization targetA checked closing argument for the source-identified open bridge.

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 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

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

Detailed research inventory

Claims, milestones, and routes in the current map

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

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma4 of 74
  • counterexample1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementCan every planar graph be drawn with integer edge lengths?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementGaussian product engineintermediate
  • retained route statementRepaired Hall surplusintermediate
  • retained route statementRelative private reserveintermediate
  • retained route statementSubstitution-spread obstructionintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureNaive shelling incidence graphreported failure
  • Research targetProve sparse factorization before shelling substitution.open
  • Research targetAbsorb exported boundary corner factors recursively.open
  • Research targetClose pentagon and degenerate insertion cases.open
  • Research targetRecursive routingsuperseded
  • Research targetPentagon insertion routesuperseded
  • Narrowed routeNaive shelling incidence graphShelling variables need not remain locally incident after pivot substitution. Relative owner routing, boundary absorption, flip transversality, and degree-at-most-five induction remain viable.
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 bridgeProve sparse factorization before shelling substitution.

1 approach has 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.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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 sparse factorization before shelling substitution.

Harborth's 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

Many local gadgets and graph classes are controlled; the unresolved step is fitting them together while preserving planarity and target neighborhoods.

  • 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 references2 cited works · next context review by Nov 21, 2026

The mathematical context was checked on Aug 21, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On the Harborth Conjecture. Part Ipreprint · Shiva Kintali · arXiv · 2026 · accessed Aug 21, 2026
  2. 2
    Harborth's Conjecture for 4-Regular Planar Graphspeer reviewed result · Daniel J. Chang, Timothy Sun · LIPIcs Graph Drawing · 2024 · accessed Aug 21, 2026

Important qualifications

  • This was a bounded status and identity check, not an exhaustive bibliography, priority review, or legal review.
  • Private packet claims were not treated as external authority; submitted links and attachments were not executed or actively rendered.
  • No absence claim is inferred from the bounded search, and recent preprints remain subject to ordinary scholarly review.

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