CERT-SUPPORT-NONOM-K7 supplies the exact drawing and matching-free rejection data. Its support-distance obstruction rules out containment, without excluding a count reduction that changes the support freely. A nonincreasing reduction with freely changing support, or a compatible neutral-component route, remains distinct from the failed containment requirement.
Route status · Narrowed routeCrossings in drawings of complete graphs
Harary–Hill conjecture
Collaboration betaHow few crossings can a drawing of a complete graph have? The source reports exact finite cases, obstructions to stronger descent rules and improved local crossing repairs, while leaving the Harary–Hill lower bound open for unrestricted drawings.
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Known results and sources 
Research problem
Exact mathematical statement
For every complete graph K_n, prove the lower bound
Here crossing number is minimized over good drawings: distinct vertices, simple edge arcs, finitely many proper crossings, no adjacent-edge or triple crossings, no tangencies or overlapping arcs, and at most one crossing between independent edges. The source reports that Hill drawings attain this value. Its unrestricted lower-bound argument remains incomplete.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Harary–Hill conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source records that each crossing survives n-4 vertex deletions. The resulting identity reduces succeeding even-order lower bounds to odd-order bounds, with the small zero-target orders handled separately.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
Harary–Hill conjecture in numbers
1.1kretained lines of mathematical investigation1,147 in the current working snapshot
- Argument development
- 710 · 62%
- Explored or eliminated routes
- 85 · 7%
- Computational analysis
- 126 · 11%
- Open obligations
- 71 · 6%
- Definitions and setup
- 155 · 14%
9selected mapped statements2routes 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.
Recommended next task
Prove global accessibility through compatible nonincreasing moves, including large convex components and interacting nonconvex cores.
Suggested move: Suggested next step: State a component potential or selection lemma for the uncovered interacting cores, tracking changed edge curves, conserved counts and goodification explicitly.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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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
CERT-SUPPORT-NONOM-K7 supplies the exact drawing and matching-free rejection data. Its support-distance obstruction rules out containment, without excluding a count reduction that changes the support freely. A nonincreasing reduction with freely changing support, or a compatible neutral-component route, remains distinct from the failed containment requirement.
Route status · Narrowed routeAll triangle degrees of that exact drawing are at most four. A neutral move can reach a matroid support at unchanged cost, while its sharp distances to any strictly improving drawable support and to cost 100 are fourteen and twenty-eight respectively. Neutral components followed by strict descent remain a possible route; the witness does not prove that every terminal component contains a matroid support.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove global accessibility through compatible nonincreasing moves, including large convex components and interacting nonconvex cores.Suggested move: Suggested next step: State a component potential or selection lemma for the uncovered interacting cores, tracking changed edge curves, conserved counts and goodification explicitly.
Ready to work on
Derive a drawing-valid rotation-energy inequality that supplies positivity beyond the identities and scalar relaxations already available.Suggested move: Suggested next step: Identify the drawing-only compatibility information used by a proposed inequality and confront it with the recorded generalized-tensor and real-relaxation obstructions.
Ready to work on
Prove the all-order common-parity extremal bound with one globally shared crossing-pair assignment.Suggested move: Suggested next step: Formulate the common witness and residual-cycle consistency conditions before combining local inequalities, and record why separately chosen local witnesses cannot substitute for them.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The unrestricted crossing-number formula for complete graphs remains open. Results for shellable drawings and recent work on shortest-geodesic drawings on the sphere address specified drawing classes; neither restriction covers every drawing in the full conjecture.
[2][3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintStreltsova and Wagner report the lower bound for shortest-geodesic drawings on the sphere through a sublevel-arrangement argument.[3] PreprintÁbrego and coauthors establish the conjectured bound for sufficiently shellable drawings, including cylindrical and other structured classes.[2] Historical sourceHarary and Hill present the complete-graph crossing construction and conjectural matching minimum; their paper also cites Guy's earlier work.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusHarary–Hill conjecture
The shellability hypothesis controls the proved drawing class; it cannot be dropped when applying the result.
[2]The 2025 preprint's spherical-arc result concerns shortest geodesic edges, not arbitrary topological arcs.
[3]The original paper allows curved edges and treats planar and spherical crossing formulations together; straight-line restrictions define a narrower task.
[1]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA proposed formalization must fix good drawings, crossings, and the complete-graph minimum before importing any restricted drawing theorem.
- Formalization targetEach shellability or spherical-arc argument needs an explicit domain bridge; no checked universal bridge was identified here.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
7 standing statements2 proposed statements3 open questions2 narrowed routes
- theorem candidate
1 of 9 1 - reduction
2 of 9 2 - lemma
3 of 9 3 - computational claim
3 of 9 3
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
- retained route statementCan every drawing of K_n be shown to have at least the Hill number of crossings?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementVertex-deletion reductionintermediate
- retained route statementTHM-PARITY-SUPPORT-9-10intermediate
- retained route statementCERT-ENDPOINT-DESCENT-K7intermediate
- retained route statementThree distances for the K11 witnessintermediate
- retained route statementTHM-NESTED-BLOCKER-REPAIRintermediate
- retained route statementTHM-ISOLATED-NONCONVEX-POPULATIONSintermediate
- 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 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
- 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 failureReducing every drawing to a contained matroid supportreported failure
- Useful failureUniversal strict single-star descentreported failure
- Research targetProve global accessibility through compatible nonincreasing moves, including large convex components and interacting nonconvex cores.open
- Research targetDerive a drawing-valid rotation-energy inequality that supplies positivity beyond the identities and scalar relaxations already available.open
- Research targetProve the all-order common-parity extremal bound with one globally shared crossing-pair assignment.open
- ComputationThe source reports a complete graph search over 952,446 root-fixed K7 rotation systems and a separate forest check for path existence.All states reportedly reach nine crossings, with maximum shortest distance nine certified by the complete graph BFS. The retained forest certifies path upper bounds, not shortestness by itself. Eight larger K11 witnesses are a seeded set, not a K11 census. · reported unreproduced
- Narrowed routeReducing every drawing to a contained matroid supportCERT-SUPPORT-NONOM-K7 supplies the exact drawing and matching-free rejection data. Its support-distance obstruction rules out containment, without excluding a count reduction that changes the support freely. A nonincreasing reduction with freely changing support, or a compatible neutral-component route, remains distinct from the failed containment requirement.
- Narrowed routeUniversal strict single-star descentAll triangle degrees of that exact drawing are at most four. A neutral move can reach a matroid support at unchanged cost, while its sharp distances to any strictly improving drawable support and to cost 100 are fourteen and twenty-eight respectively. Neutral components followed by strict descent remain a possible route; the witness does not prove that every terminal component contains a matroid support.
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
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
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
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Prepared starting pointProve global accessibility through compatible nonincreasing moves, including large convex components and interacting nonconvex cores.
Harary–Hill conjecture · ready to start
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Research contextPrepared context for any AI agent
How few crossings can a drawing of a complete graph have? The source reports exact finite cases, obstructions to stronger descent rules and improved local crossing repairs, while leaving the Harary–Hill lower bound open for unrestricted drawings.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references3 cited works · next context review by Dec 7, 2026
The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.
- 1On the Number of Crossings in a Complete Graphoriginal source · Frank Harary, Anthony Hill · Proceedings of the Edinburgh Mathematical Society · 1963 · accessed Sep 7, 2026
- 2Shellable drawings and the cylindrical crossing number of K_npreprint · Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar · arXiv · 2013-10-11 · ARXIV 1309.3665 · accessed Sep 7, 2026
- 3Sublevels in arrangements and the spherical arc crossing number of complete graphspreprint · Elizaveta Streltsova, Uli Wagner · arXiv · 2025-04-10 · ARXIV 2504.07770 · accessed Sep 7, 2026
Important qualifications
- This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
- No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
- No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
- Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
- Stronger finite-order claims appeared in a 2021 conference paper and 2025 teaching material than in an older account. The older account alone is not a contradiction. This collection did not adjudicate the stronger claim and makes no exact largest-verified-order assertion.
- The 1963 original supports Harary and Hill's published formulation, not exclusive priority; its earlier references prevent assigning a unique proposed year here.
- No computation or formal proof was checked, and recent preprints remain labeled as preprints.
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