{
  "abstract": "We study the graph $\\mathcal{AO}(G)$ of acyclic orientations of a graph $G$. Two acyclic orientations are adjacent in this graph if they disagree on the orientation of a single arc. In particular, we focus on the Hamiltonicity of the graphs $\\mathcal{AO}(G)$. Using two methods of pattern lacing which generalize the zig-zag method of Brenner, Cardinal, McConville, Merino and Mütze, we characterize which multipaths are $\\mathcal{AO}$-Hamiltonian. Moreover, we give a criterion for the gluing of a multipath on a given graph to preserve $\\mathcal{AO}$-Hamiltonicity. Building towards an inductive certification of $\\mathcal{AO}$-Hamiltonicity via the (open) ear decomposition of 2-connected graphs, we propose three ways of gluing several multipaths to a given graph. In addition, we define the acyclic polynomials to encapsulate both the number of acyclic orientations of a graph and the \"parity problem\" proposed by Savage, Squire and West: if $-1$ is not a root of the acyclic polynomial of $G$, then $G$ is not $\\mathcal{AO}$-Hamiltonian. We explore numerous properties of the acyclic polynomials, proving that they are not instances of the famous Tutte-Whitney polynomials, but that they too exhibit a partial deletion-contraction phenomenon.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02249",
  "author_total": 2,
  "authors": [
    {
      "name": "Leonie Mühlherr"
    },
    {
      "name": "Germain Poullot"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fst0x674shpt3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02249",
  "fulltext_url": null,
  "id": "fst0x674shpt3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02249",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7207747730",
  "pdf_url": "https://arxiv.org/pdf/2609.02249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fst0x674shpt3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.02249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02249"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hamiltonicity of graphs of acyclic orientations and acyclic polynomials",
  "title_text": "Hamiltonicity of graphs of acyclic orientations and acyclic polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/fst0x674shpt3",
  "venue": "arXiv",
  "versions": []
}