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CS-786 · computer science · mathematics · Graph theory and combinatorics

Duplicate Light-Tournament Two-Coloring Question

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A tournament has exactly one directed arc between every pair of distinct vertices. A vertex set is acyclic if its induced tournament has no directed cycle; two-colorability partitions all vertices into two such sets.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that this extraction repeats the complexity question represented canonically by OWR-14298804-014. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30006111. Separate source IDs are preserved for their provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
The open question is to determine the complexity of 2-coloring light tournaments: Is there a polynomial-time algorithm to color a 2-colorable light tournament with two colors? Open Problem: Complexity of 2-coloring light tournaments Alantha Newman (joint work with Felix Klingehoefer) [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The supplied fragment concerns duplicate Light-Tournament Two-Coloring Question. Its complete mathematical target is not recovered.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006112 and code OWR-14298804-015 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30006111. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-786-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-786

Duplicate Light-Tournament Two-Coloring Question

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A tournament has exactly one directed arc between every pair of distinct vertices. A vertex set is acyclic if its induced tournament has no directed cycle; two-colorability partitions all vertices into two such sets.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that this extraction repeats the complexity question represented canonically by OWR-14298804-014. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30006111. Separate source IDs are preserved for their provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
The open question is to determine the complexity of 2-coloring light tournaments: Is there a polynomial-time algorithm to color a 2-colorable light tournament with two colors? Open Problem: Complexity of 2-coloring light tournaments Alantha Newman (joint work with Felix Klingehoefer) [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The supplied fragment concerns duplicate Light-Tournament Two-Coloring Question. Its complete mathematical target is not recovered.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006112 and code OWR-14298804-015 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30006111. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

Testing first-fit on the actual light class

This repeated question concerns the same primary definition: an arc $u\to v$ is light when $\{w:v\to w\to u\}$ induces an acyclic tournament. The attempted algorithm processes vertices in a fixed order and gives each vertex the first of two colors that leaves its color class acyclic, without recoloring. Consider vertices $0,1,2,3,4,5$, with the following fifteen arcs (exactly one for each pair): \[ 01,04,05,12,20,25,30,31,32,41,42,43,51,53,54, \] where $ij$ denotes $i\to j$.
Proposition: This tournament is light and two-colorable, but first-fit in the displayed vertex order fails at vertex five.
Proof: The complete list of directed triangles, recorded as unordered triples, is \[ 012,024,034,035,125,235,245. \] This list follows by checking the twenty triples: a triple is cyclic exactly when each vertex has one outgoing arc within it. From the arc list, the neighborhoods of $01,04,05,12,20,25,30,31,32,41,42,43,51,53,54$ respectively have the following topological orders: \[ (2),(3,2),(3),(0,5),(4,1),(4,3,1),(5,4),(),(5),(),(0,5),(0),(2),(2,0),(2). \] Every earlier listed vertex points to every later listed vertex in its neighborhood. Hence all neighborhoods are acyclic and the tournament is light. First-fit places $0,1$ in color zero. Triangle $012$ forces $2$ into color one. Vertex $3$ joins color zero, since $\{0,1,3\}$ is acyclic. Vertex $4$ cannot join it because of $034$, and joins $\{2\}$ in color one. Finally $5$ creates triangle $035$ in color zero and triangle $245$ in color one, so both choices fail. Nevertheless $\{0,2\}$ and $\{1,3,4,5\}$ are acyclic: neither contains any of the seven listed triangles. They give a valid two-coloring. ∎
Thus lightness plus a two-colorability promise does not justify irreversible first-fit. The example is not a hardness proof and does not exclude recoloring or a special ordering.

Verification and remaining gaps

The example was found by enumerating labeled tournaments on five and then six vertices; the search found none on five for this fixed first-fit failure condition and found the six-vertex mask 2105. Independent direct checks verified all arc neighborhoods, all twenty triples, the failed coloring sequence and the successful partition. No global two-color algorithm is obtained.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 p. 2993, duplicated light-tournament question. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.
Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch011/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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