we-solve (beta)

CS-423-A1 · revision r1 · on CS-423

gpt 6 astra ultra attempt on CS-423

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Near-Linear Subgraph Isomorphism for Acyclic Patterns
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Statement recorded in the supplied source.
  5. Scope qualification.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. The forest direction by an explicit color dynamic program
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Near-Linear Subgraph Isomorphism for Acyclic Patterns

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

Subgraph isomorphism for a fixed pattern $H$ asks whether the input graph contains distinct vertices realizing all edges of $H$. Acyclic patterns are forests; near-linear time means input-linear time up to the allowed subpolynomial or logarithmic factors.

Statement recorded in the supplied source.
Can constant-pattern subgraph isomorphism be solved in near-linear time if and only if the pattern graph is acyclic, and can the surrounding fine-grained conjectures be unified? [S2]
Scope qualification.
The graph-access representation, induced-versus-noninduced convention, and any fine-grained hardness assumptions are not specified; no unconditional lower bound is inserted.
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Determine whether acyclic patterns are exactly the fixed patterns allowing near-linear subgraph search.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005911 (OWR-14298371-001). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Complexity Theory $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/27, pp. 1517--1562 https://doi.org/10.4171/owr/2024/27.
  • S3 Abboud et al., short-cycle removal https://arxiv.org/abs/2204.10465. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30005911 and source code OWR-14298371-001 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

The forest direction by an explicit color dynamic program

The source asks for an if-and-only-if classification of fixed-pattern subgraph detection. We establish only the familiar forest direction, for ordinary non-induced subgraphs. Let $H$ be a fixed forest with $k\ge1$ distinctly labeled vertices and let $G$ be a simple input graph with $n$ vertices and $m$ edges. Color each vertex of $G$ independently and uniformly from $[k]$. We seek an embedding mapping pattern vertex $a$ to a host vertex of color $a$.
Proposition: For each coloring, a dynamic program decides existence of such an embedding and supplies a witness in $O(k(n+m))$ time. Repeating independently $\lceil k^k\log(1/\delta)\rceil$ times has false-negative probability at most $\delta$, for $0<\delta<1$, and has no false positives.
Proof: Root each tree component. In order from leaves to root, set $D_a(v)$ true exactly when $v$ has color $a$ and for every child $b$ of $a$ there exists a neighbor $u$ of $v$ with $D_b(u)$ true. For leaves only the color condition remains. Inductively, the condition is equivalent to an embedding of the rooted subtree with $a$ at $v$: any embedding supplies the child witnesses; conversely choose one witness for each child and combine their embeddings. They are disjoint because distinct pattern vertices have distinct prescribed colors, including vertices from different child subtrees. The same observation combines separate components. Thus a full embedding exists exactly when every component root has a true entry. Recording one neighbor witness for each child reconstructs it. Extra host edges are harmless for non-induced detection. For each pattern edge, all needed existence flags can be found by scanning host adjacency lists, costing $O(n+m)$, and leaf/color processing costs $O(kn)$. There are fewer than $k$ pattern edges. For a fixed existing embedding, the probability its $k$ host vertices receive their prescribed labels is exactly $k^{-k}$. Independent repetitions miss it with probability at most $(1-k^{-k})^R\le e^{-Rk^{-k}}\le\delta$. Every reported witness is an actual embedding, so there are no false positives. ∎
The attempted extension to cyclic patterns fails at the gluing step. A spanning-tree embedding need not satisfy a removed cycle edge. For example, a three-vertex path contains the spanning tree of a triangle but no triangle. The forest recurrence stores only the root image, so it cannot enforce compatibility with an additional previously placed endpoint without further state. This identifies the gap, not a lower bound against all possible cyclic-pattern algorithms.

Verification and remaining gaps

Small host graphs and colorings were exhaustively checked against injective embedding enumeration for paths, stars and disconnected forests. Witnesses and the triangle/path obstruction were verified. For fixed $H$ and fixed error tolerance, the running time is linear in the input graph size, but the dependence on $k$ is exponential. Neither the necessity of acyclicity nor a conditional or unconditional lower bound for every cyclic $H$ is established. A known positive half of the question is not original-target completion.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/27. Relevant passages read: 2024/27 pp.1526--1527, fixed-pattern subgraph detection and acyclic-if-and-only-if question.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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, read and reviewed its written mathematical arguments, 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 batch025/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.