The source's superseded-route ledger states that a new barrier can appear and therefore demands a global potential or an explicit post-switch perfect matching. A safe switching argument remains proposed if it supplies an explicit perfect matching, a retained two-port splice, or strict descent of a well-founded global potential.
Route status · Narrowed routeGraph flows · bridgeless graphs · cubic reductions · matching obstructions
Tutte’s 5-Flow Conjecture
Collaboration betaCan every bridgeless graph carry a conserved, nonzero flow using values smaller than five? A 6-flow is known for every bridgeless graph, but the universal 5-flow conjecture remains open.
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Known results and sources 
Research problem
Exact mathematical statement
Tutte's 5-flow conjecture asks whether
Equivalently, after orienting the edges, assign nonzero values modulo five so that flow entering each vertex equals flow leaving it.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Tutte’s 5-Flow Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
In the current work's reduced cubic setup, a canonical 5-weak bisection is 5-strong exactly when its weighted deficit-token graph has a perfect matching, shifting the local obstruction to Hall barriers and switchable token zones.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
Tutte’s 5-Flow Conjecture in numbers
1.9kretained lines of mathematical investigation1,878 in the current working snapshot
- Argument development
- 1,506 · 80%
- Explored or eliminated routes
- 94 · 5%
- Computational analysis
- 90 · 5%
- Open obligations
- 24 · 1%
- Definitions and setup
- 164 · 9%
8selected mapped statements1routes investigated4open questions4contribution-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
Resolve the nonexceptional locked-interval branch with a certified port, routing certificate, short circuit, or proper deficient subset.
Suggested move: Apply the source's incoming and outgoing locks to the internal routing decomposition and retain an exact certificate for whichever outcome occurs.
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
The source's superseded-route ledger states that a new barrier can appear and therefore demands a global potential or an explicit post-switch perfect matching. A safe switching argument remains proposed if it supplies an explicit perfect matching, a retained two-port splice, or strict descent of a well-founded global potential.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Resolve the nonexceptional locked-interval branch with a certified port, routing certificate, short circuit, or proper deficient subset.Suggested move: Apply the source's incoming and outgoing locks to the internal routing decomposition and retain an exact certificate for whichever outcome occurs.
Ready to work on
Control the exceptional source-change branch and rule out cycles of successive barrier-source changes.Suggested move: Track the persistent edge cz through the recomputed three-zone decomposition, checking for a direct or neutral two-port splice before defining a descent coordinate.
Ready to work on
Global switching and descent
The first unverified step is to find a permitted move producing a perfect matching, retained splice, or strict descent from every safe deficiency-one oddness-six state.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Close the connected pair-cut terminal-graph cases without assuming a replacement barrier is absent.Suggested move: Handle the P3 and triangle cases separately, recompute every post-switch component and obstruction, and output a matching, splice, or strict potential decrease.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedRecent work on nowhere-zero flows continues to cite the 5-flow statement as a major open conjecture while using the universal 6-flow theorem as the established relaxation.[5] Peer reviewedMazzuoccolo and Steffen proved the conjectured conclusion for cubic graphs of oddness at most four.[4] Peer reviewedKochol reduced the conjecture to cyclically 6-edge-connected snarks; this is a reduction of the open problem, not a proof.[3] Peer reviewedSeymour proved that every bridgeless graph has a nowhere-zero 6-flow, leaving the one-step strengthening to 5 open.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusTutte’s 5-Flow Conjecture
Every bridgeless graph has a nowhere-zero 6-flow. This established universal bound is weaker than the conjectured 5-flow bound.
[2][5]The general conjecture can be reduced to a restricted class of highly connected cubic graphs, without thereby solving that restricted case.
[3]A nowhere-zero 5-flow is known for this special class; the result does not cover arbitrary bridgeless graphs.
[4]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal statement of nowhere-zero integer k-flows with orientation-invariant conservation.
- Formalization targetFormal graph infrastructure for bridgeless multigraphs, cubic reductions, cuts, and flow equivalences.
- Formalization targetA checked proof of the universal 5-flow statement; the established 6-flow theorem alone is insufficient.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
6 standing statements2 proposed statements4 open questions1 narrowed routes
- theorem candidate
1 of 8 1 - reduction
1 of 8 1 - lemma
3 of 8 3 - equivalence
3 of 8 3
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
- retained route statementDoes every bridgeless graph have a nowhere-zero 5-flow?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementExact bridgeless 5-flow targetintermediate
- retained route statementDeficit-token matching equivalenceintermediate
- retained route statementOddness at most four excluded conditionallyintermediate
- retained route statementSource-reported signing theoremintermediate
- retained route statementExact one-adjacency augmentation criterionintermediate
- 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
- 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 failureTreating destruction of the current Hall barrier as automatic global progressreported failure
- Research targetResolve the nonexceptional locked-interval branch with a certified port, routing certificate, short circuit, or proper deficient subset.open
- Research targetControl the exceptional source-change branch and rule out cycles of successive barrier-source changes.open
- Research targetClose the connected pair-cut terminal-graph cases without assuming a replacement barrier is absent.open
- Research targetGlobal switching and descentopen
- Narrowed routeTreating destruction of the current Hall barrier as automatic global progressThe source's superseded-route ledger states that a new barrier can appear and therefore demands a global potential or an explicit post-switch perfect matching. A safe switching argument remains proposed if it supplies an explicit perfect matching, a retained two-port splice, or strict descent of a well-founded global potential.
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
1 approach has 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 pointResolve the nonexceptional locked-interval branch with a certified port, routing certificate, short circuit, or proper deficient subset.
Tutte’s 5-Flow Conjecture · ready to start
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Research contextPrepared context for any AI agent
Can every bridgeless graph carry a conserved, nonzero flow using values smaller than five? A 6-flow is known for every bridgeless graph, but the universal 5-flow conjecture remains open.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references5 cited works · next context review by Nov 14, 2026
The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.
- 1A contribution to the theory of chromatic polynomialsoriginal source · W. T. Tutte · Canadian Journal of Mathematics 6 · 1954 · DOI 10.4153/CJM-1954-010-9 · accessed Aug 14, 2026
- 2Nowhere-zero 6-flowspeer reviewed result · Paul D. Seymour · Journal of Combinatorial Theory, Series B 30 · 1981 · DOI 10.1016/0095-8956(81)90058-7 · accessed Aug 14, 2026
- 3Reduction of the 5-flow conjecture to cyclically 6-edge-connected snarkspeer reviewed result · Martin Kochol · Journal of Combinatorial Theory, Series B 90 · 2004 · DOI 10.1016/S0095-8956(03)00080-7 · accessed Aug 14, 2026
- 4Nowhere-zero 5-flows on cubic graphs with oddness 4peer reviewed result · Giuseppe Mazzuoccolo, Eckhard Steffen · Journal of Graph Theory · 2016 · ARXIV 1412.5398 · accessed Aug 14, 2026
- 5Approximately Packing Dijoins via Nowhere-Zero Flowspeer reviewed result · Kartik Chandrasekaran, Chao Liu, R. Ravi · Combinatorica · 2025 · DOI 10.1007/s00493-025-00159-x · accessed Aug 14, 2026
Important qualifications
- The bounded pass covered original bibliographic identity, current open status, the universal 6-flow theorem, representative reductions and special cases, and recent peer-reviewed context; it was not exhaustive.
- The private packet and its URLs were not used as external authority. No submitted URL or attachment was fetched, executed, compiled, or rendered.
- No formal statement or proof of the full conjecture was identified in the bounded official-library search; absence from this search is not proof of absence.
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