{
  "abstract": "The switch Markov chain is a natural method for sampling graph realizations with a prescribed degree sequence, and $P$-stability is a general sufficient condition for its rapid mixing. Since $P$-stability is often difficult to verify directly, we develop an approach based on the simpler and more readily verifiable property of \\emph{full graphicality}. We consider bidegree sequence regions obtained by fixing the two class sizes, the common degree sum, and lower and upper bounds for the degrees in each class. Such a region is fully graphic if every bidegree sequence satisfying these constraints has a bipartite realization. Our main result shows that every fully graphic region of this form is $P$-stable. More precisely, for class sizes $n$ and $m$, the perturbation ratio is at most $(nm)^5$. It follows from a known theory that the switch Markov chain is rapidly mixing on every such region. We also show that full graphicality can be tested on a canonical extremal bidegree sequence and use this characterization to derive several new explicit sufficient conditions. By encoding loopless directed graphs as bipartite graphs, we develop a parallel theory for directed degree sequences. This gives analogous extremal characterizations, sufficient conditions for full graphicality, and $P$-stability and rapid-mixing results for directed degree-sequence regions. The encoding process allows most corresponding statements for the two cases to be formulated and proved in one run.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35777",
  "author_total": 4,
  "authors": [
    {
      "name": "Péter L. Erdős"
    },
    {
      "name": "Gábor Lippner"
    },
    {
      "name": "Na'ama Nevo"
    },
    {
      "name": "Lajos Soukup"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59xa9wmkpftch.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35777",
  "fulltext_url": null,
  "id": "59xa9wmkpftch",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35777",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7224023324",
  "pdf_url": "https://arxiv.org/pdf/2610.35777",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59xa9wmkpftch.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35777",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35777"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "P-stable degree sequence regions for bipartite and directed graphs",
  "title_text": "P-stable degree sequence regions for bipartite and directed graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/59xa9wmkpftch",
  "venue": "arXiv",
  "versions": []
}