{
  "abstract": "A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27168",
  "author_total": 1,
  "authors": [
    {
      "name": "Sergey Pupyrev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1fxhjnp1vng32.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27168",
  "fulltext_url": null,
  "id": "1fxhjnp1vng32",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27168",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214181582",
  "pdf_url": "https://arxiv.org/pdf/2609.27168",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1fxhjnp1vng32.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27168",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27168"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Smallest Cubic Non-1-Planar Graphs",
  "title_text": "Smallest Cubic Non-1-Planar Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/1fxhjnp1vng32",
  "venue": "arXiv",
  "versions": []
}