AutoGraphForge: Towards Automated Graph Theory Discovery
AutoGraphForge is an end-to-end computational pipeline for automated graph-theoretic conjecturing, refuting, formalizing, and proving, integrating symbolic and neural methods. Conjecture generation uses a counterexample-guided iterative loop with Graffiti3, a novelty filter of 559 classical/folklore relations checked via linear programming, and a refutation dataset of ~348,000 graphs. The pipeline produced 6,522 surviving conjectures, including nontrivial relations between the annihilation numbe
Analysis
TL;DR
- AutoGraphForge is an end-to-end computational pipeline for automated graph-theoretic conjecturing, refuting, formalizing, and proving, integrating symbolic and neural methods.
- Conjecture generation uses a counterexample-guided iterative loop with Graffiti3, a novelty filter of 559 classical/folklore relations checked via linear programming, and a refutation dataset of ~348,000 graphs.
- The pipeline produced 6,522 surviving conjectures, including nontrivial relations between the annihilation number and edge-cover number for bipartite and regular graphs, some of which were manually proven.
- A formalization stage deterministically translates surviving conjectures into Lean 4 statement skeletons, with kernel-verified proofs against a pinned mathlib4 and custom invariant preamble.
- Two neural provers—DeepSeek-Prover-V2-671B (vLLM-served) and Lean-specialized OProver-32B—are integrated behind the independent kernel check, with the full pipeline running on an HPC cluster.
Why It Matters
AutoGraphForge represents a significant step toward fully automated mathematical discovery in combinatorics, bridging the gap between empirical conjecture generation and machine-checked formal proof. For AI researchers, it demonstrates a practical architecture for combining symbolic reasoning, large-scale counterexample search, and neural theorem proving in a single verified pipeline. For the broader mathematics community, it shows that automated systems can produce genuinely novel, nontrivial results that survive both computational refutation and human verification.
Technical Details
- Conjecture generation loop: Graffiti3 proposes conjectures over an evolving snapshot table T (starting with a few hundred graphs), which grows only by counterexamples to its own conjectures, creating a self-improving feedback cycle.
- Novelty filtering: A filter of 559 classical and folklore graph relations, closed under transitive composition and linear identity substitution, uses linear programming to eliminate candidates already implied by known results.
- Refutation dataset: Approximately 348,000 graphs drawn from the complete House of Graphs invariant export, exhaustive census of all connected graphs on at most nine vertices, extremal families (strongly regular, minimal Ramsey, Cayley, cages, barbells, lollipops, spiders), and random graph models.
- Formalization and proving: Surviving conjectures are deterministically translated into Lean 4 statement skeletons; proofs are kernel-verified against a pinned mathlib4 and custom invariant preamble, with DeepSeek-Prover-V2-671B and OProver-32B serving as neural provers behind the independent check.
- Implementation: End-to-end pipeline running on an HPC cluster, yielding 6,522 conjectures that passed all refutation, novelty, and active-search stages.
Industry Insight
- The counterexample-guided iterative design offers a reusable template for automated discovery systems in other mathematical domains, where conjecture generation can be coupled with growing counterexample sets to progressively refine search space.
- Integrating neural provers behind a formal kernel check (Lean 4 + mathlib4) provides a robust architecture for ensuring correctness in AI-assisted theorem proving, setting a precedent for hybrid symbolic-neural verification pipelines.
- The scale of the refutation dataset (~348K graphs) and the HPC-backed iteration suggest that automated mathematical discovery is becoming computationally feasible at scale, likely accelerating the rate of novel conjecture production in combinatorics and related fields.
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