Computer-Assisted Verification of Erd\H{o}s Problem \#580 for Small Orders

This repository accompanies a computer-assisted investigation of the following extremal graph problem.

Problem.
Let (G) be a graph on (n) vertices such that at least (n/2) vertices have degree at least (n/2). Must (G) contain every tree on at most (n/2) vertices?

The accompanying work establishes the stated vertex-formulation for the finite range (1\leq n\leq 19). The result is a certified finite partial result and is not presented as a proof of the problem for all (n), nor as a verification of the stronger classical edge-formulation of the Loebl–Komlós–Sós conjecture.

The repository contains the implementation and certification material used in the investigation, including:

1. mathematical reductions of potential counterexamples to a canonical structural form;

2. enumeration and classification of all relevant tree configurations;

3. explicit witnesses supporting the classification of all 47 unlabeled trees on nine vertices;

4. reduction of the remaining exceptional trees to marked rooted-core embedding problems;

5. SAT encodings of all admissible reduced host graphs;

6. the resulting CNF instances;

7. DRUP proof certificates establishing unsatisfiability;

8. independent semantic audits of the complete CNF formulas;

9. scripts for regenerating the classification data, audit tables, and LaTeX appendix;

10. software information, cryptographic hashes, solver logs, and reproducibility instructions.

The computational verification proceeds through the following stages:

1. Reduction of any potential counterexample to a balanced host graph with a prescribed degree structure.

2. Enumeration of the relevant unlabeled trees and application of known embedding results.

3. Identification of the small set of exceptional tree configurations.

4. Reduction of each exceptional tree to a marked rooted-core embedding problem.

5. Construction of SAT formulas encoding reduced host graphs that avoid the corresponding rooted core.

6. Independent reconstruction and semantic comparison of every base clause and avoidance clause.

7. Verification of unsatisfiability using a modern SAT solver.

8. Independent checking of the associated DRUP proof certificates.

The detailed tree-classification files are generated directly by the released scripts rather than maintained manually. For every one of the 47 trees on nine vertices, the generated audit records the applicable theorem or reduction and the corresponding structural witness.

The strengthened CNF audit is implemented independently of the production generator. It reconstructs the complete base encoding and rooted-core avoidance clauses, checks their exact order and multiplicity, verifies the declared variable and clause counts, detects duplicate clauses, and compares the reconstructed formulas with the archived CNF instances.

A generation-only mode is also provided so that users may regenerate the four CNF formulas and compare them byte-for-byte with the deposited reference instances without rerunning the SAT solver or recreating the proof certificates.

The repository is intended to provide a transparent, inspectable, and reproducible framework for finite investigations of this graph-embedding problem. The documentation records the mathematical reductions, software requirements, exact commands, file organization, cryptographic hashes, and procedures required to reproduce or independently audit the computations.

AI-assistance disclosure.
OpenAI’s ChatGPT 5.6 assisted with source-code preparation, documentation, repository organization, consistency checking, and editorial revision. The author independently reviewed the mathematical arguments, program logic, generated formulas, solver outputs, and proof certificates. The mathematical claims, implementation choices, verification procedures, interpretation of the results, and final conclusions remain the sole responsibility of the author.

Version 1.0.2 revision notice.
This version supersedes the previous repository release and incorporates the following corrections and reproducibility improvements:

• the Piguet–Stein caterpillar condition is correctly stated as
  [
  \ell+c\geq \left\lfloor\frac n2\right\rfloor,
  ]
  giving the threshold (9) when (n=18);

• the trivial one-vertex-tree case is treated separately in the mathematical proof;

• the complete 47-tree classification, detailed JSON audit, and LaTeX appendix table are generated reproducibly by the released classifier;

• the CNF semantic audit has been replaced by a stronger standalone reconstruction of the complete formulas;

• a generation-only option and byte-level reference comparison have been added;

• the discussion of scope, novelty, significance, and limitations has been expanded;

• the documentation, manifests, metadata, and reproducibility instructions have been updated accordingly.

The correction to the Piguet–Stein threshold does not change the final classification. The same five exceptional trees remain, and the four CNF instances, DRUP proof certificates, unsatisfiability conclusions, and resulting finite mathematical theorem are unchanged.