{
  "abstract": "A digraph $F$ is avoidable if, for every positive integer $k$, there exists an integer $d$ such that every digraph of minimum out-degree at least $d$ contains an $F$-free subdigraph of minimum out-degree at least $k$. We prove that no digraph admitting a height function is avoidable, where a height function increases by one along every arc. This answers a question of Christoph, Janzer, Petrova, and Steiner and, together with an additional avoidance argument, determines which orientations of $C_4$ are avoidable. Motivated by a further question of Christoph, Janzer, Petrova, and Steiner, we also study Eulerian avoidability, in which the host digraph is required to have equal in-degree and out-degree at every vertex. We show that no one-directed complete bipartite digraph with nonempty parts is Eulerian-avoidable. In contrast, the orientation of $C_4$ consisting of two directed paths of length two with common endpoints is Eulerian-avoidable. Consequently, the one-directed complete bipartite orientation is the only orientation of $C_4$ that is not Eulerian-avoidable, completing the classification of $C_4$-orientations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14368",
  "author_total": 4,
  "authors": [
    {
      "name": "Hui Lei"
    },
    {
      "name": "Xiaoyi Wang"
    },
    {
      "name": "Zhijun Xu"
    },
    {
      "name": "Zhenyu Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/480byw9pyewsy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14368",
  "fulltext_url": null,
  "id": "480byw9pyewsy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14368",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213335522",
  "pdf_url": "https://arxiv.org/pdf/2609.14368",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/480byw9pyewsy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14368",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14368"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Avoidability of Digraphs with Height Functions and Orientations of $C_4$",
  "title_text": "Avoidability of Digraphs with Height Functions and Orientations of C_4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/480byw9pyewsy",
  "venue": "arXiv",
  "versions": []
}