{
  "abstract": "A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $Δ_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\\le d\\leΔ$, every finite simple $d$-degenerate graph $G$ with $Δ(G)\\leΔ$ satisfies $$q_B(G)\\le Δ+(d-1)Δ_2(G)\\le dΔ.$$ Consequently, $dΔ$ is the exact maximum, with equality precisely for graphs containing $K_{d,Δ}$. More generally, if $q_B(G)\\ge dΔ-s$, where $0\\le s<Δ$, then $G$ contains $K_{d,Δ-s}$; if also $s<d$, then $G$ has at least $d-s$ vertices of degree $Δ$ with the same open neighborhood. For $Δ\\ge3$, we further show that every $K_{3,Δ}$-free 3-degenerate graph satisfies $q_B(G)\\le3Δ-2$; the example $K_{3,Δ-1}$ shows that this bound is best possible up to one. For loopless multigraphs, we establish a sharp gap in the possible values of $q_B(G)$. For every integer $Δ\\ge3$, every finite loopless multigraph $G$ with $Δ(G)\\leΔ$ satisfies $$q_B(G)\\leΔ(Δ-1)$$ unless $G$ has a component isomorphic to $K_{Δ,Δ}$, in which case $q_B(G)=Δ^2$. The bound $Δ(Δ-1)$ is attained by both $K_{Δ,Δ-1}$ and $K_{Δ,Δ}-e$. Consequently, among finite loopless multigraphs with maximum degree at most $Δ$, no value of $q_B(G)$ lies strictly between $Δ^2-Δ$ and $Δ^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02845",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaoxue Hu"
    },
    {
      "name": "Jiangxu Kong"
    },
    {
      "name": "Yiqiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wyt5r3xy61kb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02845",
  "fulltext_url": null,
  "id": "3wyt5r3xy61kb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02845",
  "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",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7207857635",
  "pdf_url": "https://arxiv.org/pdf/2609.02845",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wyt5r3xy61kb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02845",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02845"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Degeneracy bounds, stability, and a sharp gap for $B$-colorings",
  "title_text": "Degeneracy bounds, stability, and a sharp gap for B-colorings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/3wyt5r3xy61kb",
  "venue": "arXiv",
  "versions": []
}