--- title: "Three-coloured paths in six-chromatic graphs: partial results" date: 2026-09-08 version: "0.1.0-candidate" doi: 10.5281/zenodo.22655876 pdf: https://github.com/ipitchford/three-coloured-paths-partial-results/releases/download/v0.1.0-candidate/three-coloured-paths-partial-results-0.1.0-candidate.pdf repository: https://github.com/ipitchford/three-coloured-paths-partial-results archive: https://zenodo.org/records/22655876 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Three-coloured paths in six-chromatic graphs: partial results ## Summary If a graph needs six colours on its vertices, must every three-colouring of its edges contain a one-colour path with three edges? This release does not settle that question. It supplies partial theorems, checked finite exclusions and explicit examples showing why several proposed shortcuts fail. The strongest written result handles a mixed class built from two collections of small cliques and one collection of stars. Its five-colouring theorem works at every order, under those precise hypotheses. ## Summary for specialists TCP-MIXED-001: let $G=A\cup B\cup F$ be finite and simple, where every component of $A$ and $B$ is a clique of order at most three and $F$ is a star forest. Covering edges may overlap. Then $\chi(G)\le5$. Overlap is used in the local replacement proof; the theorem does not allow arbitrary star/triangle mixtures in all three covers. For three edge-disjoint triangle factors, $\alpha(G)\ge n/3-1$ also implies five-colourability. Separately, a critical counterexample of order fourteen is excluded. The lower bound of fifteen vertices additionally assumes completeness of an external critical-graph catalogue through thirteen. At fifteen vertices, only component deficits zero and one are closed; edge counts 41, 42 and 43 remain. Here the forbidden path is an ordinary subgraph on four distinct vertices, not necessarily induced. The parent AIM-COMBINATORICS-0050 remains unresolved. ## Technical account The mixed theorem turns the two clique covers into a bipartite incidence multigraph. A minimal obstruction would have degree-five leaves. A local four-clique replacement and the classical degree-choosability theorem force their incidence graph to be a forest. Its incidence count contradicts the number of star leaves. The finite branches combine structural normalization, enumeration-coverage checks and saved vertex-colouring witnesses. A partial-Latin-square repair formulation gives a small-defect theorem, but retained counterexamples show that an arbitrary starting colouring need not admit the proposed repair. ## Evidence, assurance and limitations The package contains the full written notes, canonical claims, source dependencies, portable replay and stopped-search records. Sixteen selected checks and clean-extraction/hostile controls passed internally. Five producer-coordinated editorial roles accepted the bounded release with minor notes; this is not external specialist review or journal peer review. Catalogue completeness was not regenerated. A timeout is not an exhaustion certificate. No formal verification, historical priority or comparative research acceleration is established. The supplied review's separate audit JSON was not attached, so its reported checks do not promote an external assurance field. ## Relationship to earlier work Garrison identifies the exceptional three-colour path case. Aharoni and collaborators study the triangle-factor question and the known four-colouring bound for three star forests. The release's possible originality concerns its particular mixed theorem, reductions and obstruction objects, not those antecedent formulations. Contribution-specific precedence remains open. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Graph-colouring researchers | Audit a reusable mixed-cover theorem and finite restrictions. | Preserve cover hypotheses and external dependencies. | | Computational researchers | Reuse coverage checks and explicit repair obstructions. | Internal replay is not independent reconstruction. | | Research-method researchers | Inspect failed routes and measured scoped checks. | No matched acceleration comparator exists. | ## Why the problem matters Chromatic Ramsey questions ask how much vertex-colouring complexity forces patterns in edge-colourings. Useful partial structure can narrow the problem and prevent repeated false proof strategies without resolving the universal question. No practical impact is claimed here. ## How to inspect or reproduce the recorded checks Download the archive and read `CLAIM_RECORDS.md` before the historical notes. With Python 3.11 or later, run `replay.py --bundle-root /path/to/bundle --output-dir /path/to/fresh-output`. The output must be outside the bundle and not already exist. The wrapper checks hashes and recreates the historical layout in a copy; it does not edit the frozen evidence. Optimized Python is rejected. `test_replay.py` exercises relocation and selected hostile inputs. ## The most valuable next projects Independently scrutinize the mixed theorem's local lift and incidence count; reconstruct finite normalization coverage; assess contribution-specific prior art; and investigate an existential repair argument that survives the retained fixed-start counterexamples. These are future projects, not claims of closure. ## What is in the evidence package A consolidated PDF and accessible Markdown, all 21 original mathematical notes, canonical claim/dependency records, code, finite certificates, CNF checkpoints, replay receipts, internal editorial reports, licence map and file manifest. Externally owned papers, catalogue data and upstream binaries are omitted from the public successor with source links, hashes and reasons. The original private review archive remains unchanged. GitHub and Zenodo identify the exact public candidate version. ## Open directions for follow-up research - Resolve unrestricted mixtures of stars and triangles, including arbitrary-order triangle factors. - Resolve the remaining critical order-fifteen cases with 41, 42 or 43 edges. - Assess the mixed-cover proof and historical priority independently. - Determine whether some starting colouring always admits bipartite repair in the edge-disjoint triangle-factor subclass. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:three-coloured-paths-partial-results - Attempt and metric receipts: ep-attempt:three-coloured-paths-partial-results-remaining-assurance-publication: published / positive; scope assurance-through-publication; target Bounded partial-results candidate with fail-closed replay, five-role internal review, public immutable archives and guarded canonical readback.; active forecast 150 minutes (100-230); Fermi components Consolidated package and source checks: 1 x 30/45/65 minutes low/central/high (Large existing dossier, bounded revisions.); Internal five-role review: 1 x 20/30/50 minutes low/central/high (One differentiated round.); Public archives and media: 1 x 30/45/65 minutes low/central/high (Existing publisher tools.); Gates and public readback: 1 x 20/30/50 minutes low/central/high (Two mandatory seal/deploy loops.); positive-signal/closure probabilities 0.95/0.85 within 300 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 42/unknown/unknown/0/0/8; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -108 minutes; ratio 0.28; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No complete CPU/GPU meter across solver history, builds and hosted model calls; not inferred from wall time.; deduplicatedModelTokens: No authoritative fork-aware complete task-local total; inherited token events not summed.; uncachedInputTokens: No authoritative task-local uncached-input total.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Opening 1 retained; terminal 5 counts coordinator and four reviewer executions. (reason: Opening measurement remains immutable, not a final total.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Retain opening reworkMinutes=0; terminal approximately eight-minute recorded operational repair subtotal is disclosed as approximate. (reason: Opening snapshot is not a final total.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment - Decision object: bound — Mixed-cover colouring theorem, conditional finite exclusions and repair obstructions. Scope: Exact hypotheses in TCP-MIXED-001 and the other canonical claim records; not the full parent theorem. - Reusable methods: Structural compression (structural-compression); Certificate-first, proof-carrying research (certificate-first); Adversarial scientific controls (adversarial-controls); Productive failure and stop receipts (productive-failure); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Ordinary three-edge path avoidance gives star/triangle components; proofs, coverage and witness replay have distinct dependencies. Remaining risks: Written all-order proof awaits external specialist scrutiny.; External catalogue completeness is not regenerated.; Finite checks and timeouts do not close arbitrary orders.. - Human judgement gates: Assess the mixed theorem and local lifting argument.; Audit external theorem hypotheses and catalogue scope.; Determine contribution-specific priority.; Preserve creator, rights and candidate status. - Next assurance action: Obtain unaffiliated scrutiny of the mixed proof and independent finite-coverage reconstruction. - Claim ceiling: Unrefereed partial-results candidate. AIM-COMBINATORICS-0050 remains unresolved. The minimum-order bound depends on external catalogue completeness through thirteen; only selected order-fifteen cases are closed. No external specialist validation, formal verification or priority clearance is established. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable bounded mathematical partial results in Producer-coordinated research publication; design none; comparator No matched comparator.; estimand No speed or impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://doi.org/10.37236/5687; inherited claim: Degree-choosability characterization, Theorem 13.; inherited ceiling: Imported standard theorem; does not independently validate the new mixed-cover proof.; depends-on-claim https://users.cecs.anu.edu.au/~bdm/data/graphs.html; inherited claim: Stated edge-critical catalogue coverage through thirteen.; inherited ceiling: Supplied-record checking does not regenerate catalogue completeness. ## Verification status Unrefereed partial-results candidate. AIM-COMBINATORICS-0050 remains unresolved. The minimum-order bound depends on external catalogue completeness through thirteen; only selected order-fifteen cases are closed. No external specialist validation, formal verification or priority clearance is established. ## References 1. Garrison (2015), Good Graph Hunting, arXiv:1508.01833; exceptional three-colour path case. 2. Aharoni et al. (2018), Ramsey-nice families of graphs; triangle factors and known galaxy bounds. 3. Holliday, Vandenbussche and Westlund (2016), Theorem 13: degree-choosability characterization. 4. Gao and Postle (2019), On the minimal edge density of K4-free 6-critical graphs. 5. McKay-hosted, Lalonde-generated edge-critical catalogue; completeness remains external.