Press release · 8 September 2026 · version 0.1.0-candidate
Three-coloured paths in six-chromatic graphs: partial results
A five-colouring theorem for a mixed cover class, conditional finite exclusions and explicit repair obstructions; the parent problem remains unresolved.
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.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- 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.
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.
Cite
BibTeX
@misc{threecolouredpathspartialresults2026,
title = {Three-coloured paths in six-chromatic graphs: partial results},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22655876},
url = {https://doi.org/10.5281/zenodo.22655876},
version = {0.1.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/three-coloured-paths-partial-results/}
}Also: cite.bib · paper.json · this page as Markdown