[#P2874] Five-colouring triangle-free graphs of maximum degree six

A flat mathematical diagram showing a triangle-free graph with five vertex colors. — A schematic view of a triangle-free graph with five vertex colors.

Problem. Is every finite simple triangle-free graph \(G\) with maximum degree \(\Delta(G)\le 6\) properly colourable with at most five colours?

1Context

This is a sharply parameterized first unresolved case of a general chromatic bound. Reducible configurations, forbidden local structures, and exhaustive lower bounds on counterexample order can be reused across proof attempts.

2Problem setup

Definition 1 (A graph). A graph is triangle-free when it has no three vertices that are pairwise adjacent.

Definition 2 (A proper five-colouring). A proper five-colouring is a map from the vertex set to a set of five colours such that adjacent vertices receive different colours.

Remark 1. This is a sharply parameterized first unresolved case of a general chromatic bound. Reducible configurations, forbidden local structures, and exhaustive lower bounds on counterexample order can be reused across proof attempts.

3What counts as a solution

Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.
A counterexample must include an adjacency list plus independently checkable certificates of triangle-freeness, maximum degree, and non-five-colourability.

1Status

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved. Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.^[1]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved. Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.^[1] Relevance to this problem Defines the dated frontier and the exact remainder that research on this problem must resolve. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved. A complete resolution must satisfy this condition: Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six. Open full record →
	Trace	Reported
attempt · Route 1 Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.^[1] Relevance to this problem For Five-colouring triangle-free graphs of maximum degree six, record triangle-free-degree-six-five-colouring-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 37923; all comments concern fractional colouring or possible approaches rather than a resolution.
Abrishami and Erfanian, Discrete Mathematics 346 (2023), 113609, prove Reed's bound for maximal triangle-free graphs with maximum degree below 7 and for additional bounded-order cases; their conclusion does not cover every triangle-free graph of maximum degree 6.
Goedgebeur's arXiv:1707.07581 computationally excludes small triangle-free 6-chromatic graphs through a substantial order range, giving reusable lower bounds on any counterexample rather than a full proof.
A TheoremDB search for the exact degree-six, triangle-free, five-colouring target and Reed-conjecture aliases found no duplicate.

Recorded example 1. The condition is sharp enough to include 5-chromatic triangle-free graphs, while Brooks' theorem alone gives only a six-colour bound at maximum degree six.

Computational notes

Goedgebeur proved by exhaustive generation that the smallest triangle-free 6-chromatic graph has at least 32 vertices, but that result does not impose maximum degree six on all larger candidates.

How the 2 records connectTyped relations and evidence flow

ProblemFive-colouring triangle-free graphs of maximum degree six

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Cycle Double Cover Conjecturegraph theory
Is there a truly subcubic algorithm for weighted APSP?graph theory
The Total Coloring Conjecturegraph theory

How to cite

TheoremDB contributors, “Five-colouring triangle-free graphs of maximum degree six,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/triangle-free-degree-six-five-colouring

This page as plain text: triangle-free-degree-six-five-colouring.md

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1References

Packet source. MathOverflow: Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?. Question 37923 and all visible comments, checked through the Stack Exchange API on 2026-07-27. Question 37923 and all visible comments, checked through the Stack Exchange API on 2026-07-27. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Five-colouring triangle-free graphs of maximum degree six: UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.Source named by the research packet.
Gholamreza Abrishami and Ahmad Erfanian, “A note on Reed's conjecture for triangle-free graphs”. Discrete Mathematics 346(12) (2023), 113609. DOI 10.1016/j.disc.2023.113609. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.Also cited at Full journal article relevant to Five-colouring triangle-free graphs of maximum degree six.Source used to assess the problem's recorded status.For Five-colouring triangle-free graphs of maximum degree six: UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.
MathOverflow: Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1707.07581, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.Also cited at Full preprint relevant to Five-colouring triangle-free graphs of maximum degree six.Source used to assess the problem's recorded status.For Five-colouring triangle-free graphs of maximum degree six: UNKNOWN as of 2026-07-27. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

[#P2874] Five-colouring triangle-free graphs of maximum degree six

1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance