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Resolving Goldreich's Open Problem 4.1: Linear Lower Bounds for Cubic Graph Isomorphism Testing

Scott Schubr

Luminary Oddities

  • Computer Science & Software Systems
  • Algorithms and Complexity
  • Combinatorics and Graph Theory
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
6 · Permanent link
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Overview

Abstract

We resolve Goldreich's Open Problem 4.1 negatively: testing isomorphism to a fixed cubic graph can require linearly many queries at the specified proximity 0.01. For every sufficiently large even N, there exists a known connected simple 3-regular N-vertex target for which every randomized two-sided tester requires Ω(N) incidence queries, even when inputs are promised connected and cubic. Reading the whole incidence representation gives a matching O(N) upper bound, so the worst-case query complexity is Θ(N). Distance is measured by unordered edge changes with far threshold 0.01(3N/2); computation and target preprocessing are unrestricted.

The proof develops a cubic encoding within Dror's sparse-code framework. Its central estimate is dist_H(b,im A)≤(2/3)dist_iso(G_b,G_0) for a 16n-vertex encoding under arbitrary uncolored bijections, including equal isolate padding. Adaptive target-middle colors, a joint row-defect budget and median rounding establish the estimate. Bounded equivariant surgeries produce connected cubic targets at every sufficiently large even order. The same encoding covers every sufficiently large integer order at proximity 0.01 with at most 47 isolates, or at most one isolate using the surgeries. This also attains the minimum possible maximum-degree threshold for linear fixed-target isomorphism-testing hardness.

Keywords graph isomorphism testing · bounded-degree graphs · incidence queries · connected cubic graphs · subcubic graphs · query complexity · robust decoding

Scope of the work

A negative answer to Goldreich's Open Problem 4.1: tight linear worst-case query complexity at proximity 0.01 for fixed-target isomorphism testing on connected simple cubic graphs, with extensions to all sufficiently large orders at maximum degree three.

Reproduction notes

The 11-page manuscript and complete editable LaTeX sources are supplied in this 21-member package. README.md gives the build and reproduction commands. Reproducibility code: verify_all_orders.py, verify_combinatorics.py, verify_connectivity.py, verify_core_transfer.py, verify_cubic.py, verify_encoding.py, verify_literal.py, verify_proximity.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.

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Author disclosures

AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.

References and related publications

  1. A. Dror, A Linear Lower Bound on Testing Graph Isomorphism in the Bounded-Degree Model, ECCC TR26-268 (2026).
  2. O. Goldreich and L. Tauber, Testing in the Bounded-Degree Graph Model with Degree Bound Two, LNCS 15700 (2025), 373–382.
  3. O. Goldreich, Testing Isomorphism in the Bounded-Degree Graph Model, LNCS 15700 (2025), 293–323; Open Problem 4.1.
  4. O. Goldreich and L. Tauber, On Testing Isomorphism to a Fixed Graph in the Bounded-Degree Graph Model, LNCS 15700 (2025), 324–337.
  5. J.-Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica 12 (1992), 389–410.
  6. V. Arvind, J. Köbler, S. Kuhnert and Y. Vasudev, Approximate Graph Isomorphism, MFCS 2012, LNCS 7464, 100–111.
  7. R. O’Donnell, J. Wright, C. Wu and Y. Zhou, Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs, SODA 2014, 1659–1677.
  8. Y. Chen, J. Flum and M. Liu, Some remarks on the uncolored versions of the original CFI-graphs, arXiv:2507.01459 (2025).
  9. M. Goetze, P. Jungeblut and T. Ueckerdt, Recognition Complexity of Subgraphs of k-Connected Planar Cubic Graphs, Algorithmica 88 (2026), article 11.
  10. I. Adler, N. Köhler and P. Peng, On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing, SIAM J. Comput. 53(4) (2024), 825–883.
  11. K. Makarychev, R. Manokaran and M. Sviridenko, Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm, arXiv:1403.7721 (2014).

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Cite this edition

Scott Schubr. (2026). Resolving Goldreich's Open Problem 4.1: Linear Lower Bounds for Cubic Graph Isomorphism Testing (Version 6). Luminary Research Archive. https://research.luminaryoddities.com/linear-query-lower-bound-graph-isomorphism-testing-maximum-degree-three/v/6

BibTeX
@techreport{luminary_linear_query_lower_bound_graph_isomorphism_testing_maximum_degree_three_v6,
  title = {Resolving Goldreich's Open Problem 4.1: Linear Lower Bounds for Cubic Graph Isomorphism Testing},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 6},
  url = {https://research.luminaryoddities.com/linear-query-lower-bound-graph-isomorphism-testing-maximum-degree-three/v/6}
}

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