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The cycle visits distinct vertices, and ⌈n/r⌉ is rounded up.","source_entry_ids":["caccetta-haggkvist:entry:0:v2","caccetta-haggkvist:entry:rainbow:v1"],"formal_declarations":[]},{"statement":"For each r in {2, 3, 4, 5}, every finite loopless simple digraph with n vertices and minimum out-degree at least r contains a directed cycle of length at most ⌈n/r⌉.","id":"claim-ch-small-outdegree","slug":"ch-small-outdegree","version":2,"kind":"theorem","title":"The sharp bound for out-degree two through five","assumptions":["The degree parameter is restricted to two through five; the graph need not be regular."],"scope":"The degree parameter is restricted to two through five; the graph need not be regular.","source_entry_ids":["caccetta-haggkvist:entry:small-degree:v2"],"formal_declarations":[]},{"statement":"Every finite loopless vertex-transitive simple digraph with n vertices and minimum out-degree at least r ≥ 1 contains a directed cycle of length at most ⌈n/r⌉.","id":"claim-ch-vertex-transitive","slug":"ch-vertex-transitive","version":2,"kind":"theorem","title":"The sharp bound for vertex-transitive graphs","assumptions":["Vertex-transitivity requires the automorphism group to act transitively on vertices."],"scope":"Vertex-transitivity requires the automorphism group to act transitively on vertices.","source_entry_ids":["caccetta-haggkvist:entry:transitive:v2"],"formal_declarations":[]},{"statement":"Every finite loopless simple digraph with n vertices, minimum out-degree at least r ≥ 1 and n ≥ 2r² − 3r + 1 contains a directed cycle of length at most ⌈n/r⌉.","id":"claim-ch-large-order","slug":"ch-large-order","version":2,"kind":"theorem","title":"The sharp bound when n ≥ 2r² − 3r + 1","assumptions":["Shen’s theorem restricts the order relative to the minimum-degree parameter."],"scope":"Shen’s theorem restricts the order relative to the minimum-degree parameter.","source_entry_ids":["caccetta-haggkvist:entry:large-order:v2"],"formal_declarations":[]},{"statement":"Every finite oriented graph on n ≥ 1 vertices with minimum out-degree at least n/3 contains a directed triangle.","id":"claim-ch-triangle-target","slug":"ch-triangle-target","version":1,"kind":"conjecture","title":"The one-third directed triangle question","assumptions":["Oriented graphs have no loops, parallel arcs or opposite arcs; this is a specialization of the full conjecture."],"scope":"Oriented graphs have no loops, parallel arcs or opposite arcs; this is a specialization of the full conjecture.","source_entry_ids":["caccetta-haggkvist:entry:triangle-target:v1","caccetta-haggkvist:entry:rainbow:v1"],"formal_declarations":[]},{"statement":"Every finite oriented graph on n ≥ 1 vertices with minimum out-degree at least 0.3465n contains a directed triangle.","id":"claim-ch-triangle-03465","slug":"ch-triangle-03465","version":1,"kind":"bound","title":"A proved 0.3465 triangle threshold","assumptions":["The assumption is stronger than n/3. This published threshold is selected evidence, not a claim to be the best current bound."],"scope":"The assumption is stronger than n/3. This published threshold is selected evidence, not a claim to be the best current bound.","source_entry_ids":["caccetta-haggkvist:entry:triangle-bound:v1"],"formal_declarations":[]},{"statement":"Every simple undirected graph on n ≥ 1 vertices whose edges are partitioned into n color classes of size at least an integer r ≥ 1 contains a rainbow cycle of length at most ⌈n/r⌉.","id":"claim-ch-rainbow-cycle","slug":"ch-rainbow-cycle","version":1,"kind":"conjecture","title":"Aharoni’s rainbow cycle conjecture","assumptions":["A rainbow cycle uses pairwise distinct edge colors; this is a stronger conjecture in a different graph setting."],"scope":"A rainbow cycle uses pairwise distinct edge colors; this is a stronger conjecture in a different graph setting.","source_entry_ids":["caccetta-haggkvist:entry:rainbow:v1"],"formal_declarations":[]},{"statement":"If Aharoni’s rainbow cycle conjecture holds for all n and r, then the Caccetta–Häggkvist directed cycle bound holds for all n and r.","id":"claim-ch-rainbow-implies-directed","slug":"ch-rainbow-implies-directed","version":1,"kind":"theorem","title":"Why the rainbow conjecture would imply the directed bound","assumptions":["Handle an opposite-arc cycle first; otherwise forget directions and color each edge by its tail. Distinct colors on a cycle force one outgoing edge at each cycle vertex."],"scope":"Handle an opposite-arc cycle first; otherwise forget directions and color each edge by its tail. Distinct colors on a cycle force one outgoing edge at each cycle vertex.","source_entry_ids":["caccetta-haggkvist:entry:rainbow:v1"],"formal_declarations":[]},{"statement":"For every fixed integer r ≥ 1 there is a constant α_r, independent of n, such that every simple n-vertex edge-colored graph with n color classes of size at least r has a rainbow cycle of length at most n/r + α_r; the constants can be chosen with α_r = O(r⁵ log² r) as r grows.","id":"claim-ch-rainbow-additive","slug":"ch-rainbow-additive","version":1,"kind":"bound","title":"An additive bound for rainbow cycles","assumptions":["The constant depends on r and is not removed by this theorem."],"scope":"The constant depends on r and is not removed by this theorem.","source_entry_ids":["caccetta-haggkvist:entry:rainbow-additive:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"The full cycle bound is reported open in the selected 2025 literature. Sharp results cover some degree parameters and symmetric graphs; the triangle and rainbow bounds have separate limitations.","report_ids":["caccetta-haggkvist:report:current-open:v1"],"progress_claim_ids":["claim-ch-small-outdegree","claim-ch-vertex-transitive","claim-ch-large-order","claim-ch-triangle-03465"],"remaining":"Prove the sharp ceiling bound for all finite loopless simple digraphs and every positive integer degree parameter, without the restrictions or additive errors in the selected results."},"statement_kind":"conjecture","problem_id":"problem-caccetta-haggkvist","slug":"caccetta-haggkvist","title":"Caccetta–Häggkvist cycle bound","question":"Must a simple directed graph with n vertices and minimum out-degree at least r ≥ 1 contain a directed cycle of length at most ⌈n/r⌉?","scope":"Finite directed graphs with no loops or parallel arcs; opposite arcs are allowed and already give a cycle of length two. Here n is the number of vertices and r is a positive integer lower bound on every vertex’s out-degree. The cycle visits distinct vertices, and ⌈n/r⌉ is rounded up.","aliases":["Caccetta–Häggkvist conjecture"],"fields":["Combinatorics"],"msc":["05C20","05C38"],"topics":["Directed cycles","Minimum out-degree","Rainbow cycles"],"background":"graduate","scale":"focused question","depth":"maintained","source_entry_ids":["caccetta-haggkvist:entry:0:v2","caccetta-haggkvist:entry:small-degree:v2","caccetta-haggkvist:entry:transitive:v2","caccetta-haggkvist:entry:large-order:v2","caccetta-haggkvist:entry:triangle-bound:v1","caccetta-haggkvist:entry:triangle-target:v1","caccetta-haggkvist:entry:rainbow:v1","caccetta-haggkvist:entry:rainbow-additive:v1"],"status_report_ids":["caccetta-haggkvist:report:0:v2","caccetta-haggkvist:report:current-open:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"open","scope":"The canonical question and cited source selection only.","rationale":"The full cycle bound is reported open in the selected 2025 literature. Sharp results cover some degree parameters and symmetric graphs; the triangle and rainbow bounds have separate limitations.","decided_on":"2026-10-12","review_due_on":"2027-01-12","report_ids":["caccetta-haggkvist:report:current-open:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"Counting flags in triangle-free digraphs","url":"https://arxiv.org/abs/0908.2791v4","note":"See the cited passages and source metadata below for exact scope and revision."},{"title":"Aharoni’s rainbow cycle conjecture holds up to an additive constant","url":"https://arxiv.org/abs/2212.05697v4","note":"See the cited passages and source metadata below for exact scope and revision."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"caccetta-haggkvist-degree","title":"Minimum out-degree is a condition at every vertex","summary":"An average degree condition is weaker. A two-way pair of arcs already supplies a short cycle; the triangle question therefore uses oriented graphs.","review":"pending","kind":"interpretation","source_entry_ids":["caccetta-haggkvist:entry:triangle-target:v1","caccetta-haggkvist:entry:0:v2"],"claims":[{"id":"claim-ch-cycle-bound","version":2},{"id":"claim-ch-triangle-target","version":1}],"limitation":"The precise assumptions on each Claim determine which graph classes it covers."},{"id":"caccetta-haggkvist-flags","title":"Flag algebras and directed triangles","summary":"Hladký, Král’ and Norin combine local configuration inequalities and induction to prove the selected triangle threshold.","review":"pending","kind":"approach","source_entry_ids":["caccetta-haggkvist:entry:triangle-bound:v1"],"claims":[{"id":"claim-ch-triangle-03465","version":1}],"limitation":"The auxiliary arithmetic has not been rerun by OpenMath; the threshold exceeds one third."},{"id":"caccetta-haggkvist-rainbow-route","title":"A stronger rainbow formulation","summary":"The tail-coloring reduction connects directed cycles to rainbow cycles. The additive theorem gives a result for the stronger formulation.","review":"pending","kind":"approach","source_entry_ids":["caccetta-haggkvist:entry:rainbow:v1","caccetta-haggkvist:entry:rainbow-additive:v1"],"claims":[{"id":"claim-ch-rainbow-cycle","version":1},{"id":"claim-ch-rainbow-implies-directed","version":1},{"id":"claim-ch-rainbow-additive","version":1}],"limitation":"An additive error remains, and the exact rainbow conjecture is still open."},{"id":"caccetta-haggkvist-rainbow-impact","title":"A sharp rainbow theorem would settle the directed bound","summary":"The proved reduction transfers a resolution of the full rainbow conjecture to the directed problem.","review":"pending","kind":"impact","source_entry_ids":["caccetta-haggkvist:entry:rainbow:v1"],"claims":[{"id":"claim-ch-rainbow-implies-directed","version":1}],"scenario":{"id":"claim-ch-rainbow-cycle","version":1},"limitation":"The implication does not establish the unresolved premise."}],"coverage_gaps":["Selected sources and milestones; this is not an exhaustive literature review.","Independent mathematical review and OpenMath reproduction of external proof checks remain pending."],"checked_on":"2026-10-21"},"sources":[{"work":{"authors":["Jan Hladký","Daniel Král’","Sergey Norin"],"kind":"research_paper","publication":"published","publication_note":"Source-backed selection; publication and proof assessment are separate.","ai_role":"not_reported","ai_note":"AI use is not reported in the selected source.","artifacts":[{"kind":"project_notes","title":"Companion Maple arithmetic","url":"https://arxiv.org/src/0908.2791v4/anc/CH.mw","version":"0908.2791v4","assessment":"not_executed","note":"Companion arithmetic for the published bound; no OpenMath execution is attached."}],"published":{"value":"2017"}},"id":"hkn-triangle-bound@arxiv-0908.2791v4","source_id":"hkn-triangle-bound","title":"Counting flags in triangle-free digraphs","owner":"Jan Hladký; Daniel Král’; Sergey Norin","url":"https://arxiv.org/abs/0908.2791v4","subject_scope":"Counting flags in triangle-free digraphs","revision":"arxiv-0908.2791v4","content_sha256":null,"update_method":"Manual source review; publish new observations with immutable versions","identifier_policy":"Source revision plus exact passage or declaration; no fuzzy identity merge","observed_on":"2026-10-12","source_updated_on":null,"rights":"metadata-link","license":"Metadata, links and original OpenMath paraphrases only","terms_url":"https://arxiv.org/abs/0908.2791v4","attribution":"Jan Hladký; Daniel Král’; Sergey Norin","rights_note":"Follow the original source for authoritative statements and reuse terms.","check_interval_days":90},{"work":{"authors":["Patrick Hompe","Tony Huynh"],"kind":"research_paper","publication":"published","publication_note":"Source-backed selection; publication and proof assessment are separate.","ai_role":"not_reported","ai_note":"AI use is not reported in the selected source.","artifacts":[],"published":{"value":"2025-05"}},"id":"hompe-huynh-rainbow@arxiv-2212.05697v4","source_id":"hompe-huynh-rainbow","title":"Aharoni’s rainbow cycle conjecture holds up to an additive constant","owner":"Patrick Hompe; Tony Huynh","url":"https://arxiv.org/abs/2212.05697v4","subject_scope":"Aharoni’s rainbow cycle conjecture holds up to an additive constant","revision":"arxiv-2212.05697v4","content_sha256":null,"update_method":"Manual source review; publish new observations with immutable versions","identifier_policy":"Source revision plus exact passage or declaration; no fuzzy identity merge","observed_on":"2026-10-12","source_updated_on":null,"rights":"metadata-link","license":"Metadata, links and original OpenMath paraphrases only","terms_url":"https://arxiv.org/abs/2212.05697v4","attribution":"Patrick Hompe; Tony Huynh","rights_note":"Follow the original source for authoritative statements and reuse terms.","check_interval_days":90},{"id":"aim-caccetta@2006-04-14-sha256-98251ac3","source_id":"aim-caccetta","title":"AIM · Caccetta–Häggkvist workshop survey","owner":"Blair D. 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The paper forbids opposite arcs and uses flag algebras with auxiliary arithmetic.","wording_kind":"paraphrase","locator":"Theorem 1.2; §2 notation; §5 computation","supersedes":null},{"id":"caccetta-haggkvist:entry:triangle-target:v1","source_version_id":"hkn-triangle-bound@arxiv-0908.2791v4","item_id":"§1, triangle specialization","url":"https://arxiv.org/abs/0908.2791v4","wording":"The triangle specialization asks for a directed triangle at minimum out-degree at least one third of the vertex count, in an oriented graph.","wording_kind":"paraphrase","locator":"§1, triangle specialization","supersedes":null},{"id":"caccetta-haggkvist:entry:rainbow:v1","source_version_id":"hompe-huynh-rainbow@arxiv-2212.05697v4","item_id":"§1, Conjectures 1.1 and 1.3; reduction following Conjecture 1.3","url":"https://arxiv.org/abs/2212.05697v4","wording":"The rainbow conjecture asks for a cycle with distinct edge colors and length at most ⌈n/r⌉ when n color classes in a simple n-vertex graph each have at least r edges. Coloring directed edges by their tails shows this would imply the Caccetta–Häggkvist conjecture.","wording_kind":"paraphrase","locator":"§1, Conjectures 1.1 and 1.3; reduction following Conjecture 1.3","supersedes":null},{"id":"caccetta-haggkvist:entry:rainbow-additive:v1","source_version_id":"hompe-huynh-rainbow@arxiv-2212.05697v4","item_id":"Corollary 1.8; r = 1 discussed in its proof","url":"https://arxiv.org/abs/2212.05697v4","wording":"For each fixed positive integer r, the authors prove a rainbow cycle bound n/r + α_r, with a constant depending on r and growth O(r⁵ log² r). This leaves the sharp ceiling bound unresolved.","wording_kind":"paraphrase","locator":"Corollary 1.8; r = 1 discussed in its proof","supersedes":null}],"reports":[{"id":"caccetta-haggkvist:report:0:v2","entry_id":"caccetta-haggkvist:entry:0:v2","state":"unknown","source_label":"Conjecture 2.1; no current-status report","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":"2006-04-14","observed_on":"2026-10-21","evidence_url":"https://aimath.org/WWN/caccetta/caccetta.pdf","supersedes":"caccetta-haggkvist:report:0:v1"},{"id":"caccetta-haggkvist:report:current-open:v1","entry_id":"caccetta-haggkvist:entry:rainbow:v1","state":"reported_open","source_label":"Open question in cited source","scope":"The published 2025 paper presents the full directed cycle bound as an open conjecture; partial results below retain their own assumptions.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/2212.05697v4","supersedes":"caccetta-haggkvist:report:0:v1"},{"id":"caccetta-haggkvist:report:ch-small-outdegree:v2","entry_id":"caccetta-haggkvist:entry:small-degree:v2","state":"resolved","source_label":"Established result in cited source","scope":"The degree parameter is restricted to two through five; the graph need not be regular.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://aimath.org/WWN/caccetta/caccetta.pdf","supersedes":"caccetta-haggkvist:report:ch-small-outdegree:v1"},{"id":"caccetta-haggkvist:report:ch-vertex-transitive:v2","entry_id":"caccetta-haggkvist:entry:transitive:v2","state":"resolved","source_label":"Established result in cited source","scope":"Vertex-transitivity requires the automorphism group to act transitively on vertices.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://aimath.org/WWN/caccetta/caccetta.pdf","supersedes":"caccetta-haggkvist:report:ch-vertex-transitive:v1"},{"id":"caccetta-haggkvist:report:ch-large-order:v2","entry_id":"caccetta-haggkvist:entry:large-order:v2","state":"resolved","source_label":"Established result in cited source","scope":"Shen’s theorem restricts the order relative to the minimum-degree parameter.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://aimath.org/WWN/caccetta/caccetta.pdf","supersedes":"caccetta-haggkvist:report:ch-large-order:v1"},{"id":"caccetta-haggkvist:report:ch-triangle-target:v1","entry_id":"caccetta-haggkvist:entry:triangle-target:v1","state":"reported_open","source_label":"Open question in cited source","scope":"Oriented graphs have no loops, parallel arcs or opposite arcs; this is a specialization of the full conjecture.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/0908.2791v4","supersedes":null},{"id":"caccetta-haggkvist:report:ch-triangle-03465:v1","entry_id":"caccetta-haggkvist:entry:triangle-bound:v1","state":"resolved","source_label":"Established result in cited source","scope":"The assumption is stronger than n/3. This published threshold is selected evidence, not a claim to be the best current bound.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/0908.2791v4","supersedes":null},{"id":"caccetta-haggkvist:report:ch-rainbow-cycle:v1","entry_id":"caccetta-haggkvist:entry:rainbow:v1","state":"reported_open","source_label":"Open question in cited source","scope":"A rainbow cycle uses pairwise distinct edge colors; this is a stronger conjecture in a different graph setting.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/2212.05697v4","supersedes":null},{"id":"caccetta-haggkvist:report:ch-rainbow-implies-directed:v1","entry_id":"caccetta-haggkvist:entry:rainbow:v1","state":"resolved","source_label":"Established result in cited source","scope":"Handle an opposite-arc cycle first; otherwise forget directions and color each edge by its tail. Distinct colors on a cycle force one outgoing edge at each cycle vertex.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/2212.05697v4","supersedes":null},{"id":"caccetta-haggkvist:report:ch-rainbow-additive:v1","entry_id":"caccetta-haggkvist:entry:rainbow-additive:v1","state":"resolved","source_label":"Established result in cited source","scope":"The constant depends on r and is not removed by this theorem.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/2212.05697v4","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"The full cycle bound is reported open in the selected 2025 literature. 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This leaves the sharp ceiling bound unresolved.","wording_kind":"paraphrase","locator":"Corollary 1.8; r = 1 discussed in its proof","supersedes":null}],"sources":[{"work":{"authors":["Patrick Hompe","Tony Huynh"],"kind":"research_paper","publication":"published","publication_note":"Source-backed selection; publication and proof assessment are separate.","ai_role":"not_reported","ai_note":"AI use is not reported in the selected source.","artifacts":[],"published":{"value":"2025-05"}},"id":"hompe-huynh-rainbow@arxiv-2212.05697v4","source_id":"hompe-huynh-rainbow","title":"Aharoni’s rainbow cycle conjecture holds up to an additive constant","owner":"Patrick Hompe; Tony Huynh","url":"https://arxiv.org/abs/2212.05697v4","subject_scope":"Aharoni’s rainbow cycle conjecture holds up to an additive constant","revision":"arxiv-2212.05697v4","content_sha256":null,"update_method":"Manual source review; publish new observations with immutable versions","identifier_policy":"Source revision plus exact passage or declaration; no fuzzy identity merge","observed_on":"2026-10-12","source_updated_on":null,"rights":"metadata-link","license":"Metadata, links and original OpenMath paraphrases only","terms_url":"https://arxiv.org/abs/2212.05697v4","attribution":"Patrick Hompe; Tony Huynh","rights_note":"Follow the original source for authoritative statements and reuse terms.","check_interval_days":90}]}}}