{
  "abstract": "Every graph obtained from a regular Class $1$ graph by a vertex-splitting is overfull and hence is of Class $2$; the central question is whether the resulting graph is edge-chromatic critical. We consider a revised version of the Hilton-Zhao vertex-splitting conjecture for triangle-free base graphs, excluding the known cubic exception that arises from the Wagner graph. Let $G$ be a connected $n$-vertex triangle-free $Δ$-regular Class $1$ graph, and let $G^{*}$ be obtained from $G$ by splitting one vertex into two adjacent vertices. We first show that any non-extreme vertex-splitting produces a $Δ$-critical graph provided $Δ>\\frac{n-1}{3}$. Consequently, for non-extreme vertex-splittings, the revised version of the conjecture is valid across its entire conjectured degree range, and indeed even under a slightly weaker degree condition. We next turn to arbitrary vertex-splittings, extreme ones included. It is shown that $G^*$ is $Δ$-critical provided $n \\ge 10$ and $Δ>\\frac{2n-3}{5}$. In addition, for $n>24$, the non-strict condition $Δ\\ge \\frac{2n-3}{5}$ is already sufficient. This extends the bipartite range $Δ>\\frac{2n}{5}$ to the boundary range that the bipartite argument does not cover. In particular, our arguments give a complete treatment of the case of triangle-free $6$-regular base graphs in the conjectured range. The proofs bring together structural properties of full-deficiency pairs, homomorphism theorems for dense triangle-free graphs, and Hamilton decompositions built via graph detachment.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41984",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuli Qi"
    },
    {
      "name": "Xin Tian"
    },
    {
      "name": "Gregory Gutin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hgszf9jj00t2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41984",
  "fulltext_url": null,
  "id": "3hgszf9jj00t2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41984",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41984",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hgszf9jj00t2.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41984",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41984"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hilton-Zhao vertex-splitting conjecture for triangle-free graphs",
  "title_text": "Hilton-Zhao vertex-splitting conjecture for triangle-free graphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/3hgszf9jj00t2",
  "venue": "arXiv",
  "versions": []
}