{
  "abstract": "Let $\\mathcal{F}$ be a nonempty family of connected bipartite graphs, each with at least two edges. For a graph $G$, a proper vertex coloring of $G$ is $(2,\\mathcal{F})$-avoiding if no member of $\\mathcal{F}$ occurs bichromatically, and $χ_{2,\\mathcal{F}}(G)$ denotes the minimum number of colors in such a coloring. A B-coloring of $G$ is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in a B-coloring of $G$. For a fixed connected bipartite graph $F$ with at least one edge and bipartition classes $X_F$ and $Y_F$, define $k(F)=\\min\\{|I|:I\\subseteq X_F\\text{ or }I\\subseteq Y_F,F-I\\text{ is a forest}\\}$. Let $m\\ge2$ be the minimum number of edges in a member of $\\mathcal{F}$. We prove that if $k(F)\\le m-2$, then every $F$-free graph $G$ of sufficiently large maximum degree $Δ$ satisfies $χ_{2,\\mathcal{F}}(G)=O((\\frac{Δ^m}{\\logΔ})^{\\frac{1}{m-1}})$, which gives a positive answer to Chuet's Problem A and C in a sharp sense, thereby extending the results of Chuet [arXiv:2603.23379] from frugal colorings to $(2,\\mathcal{F})$-avoiding colorings. For B-colorings, put $k=k(F)$, $h=|V(F)|$, and $s=\\min\\{|X_F|,|Y_F|\\}$. We prove that every $F$-free graph $G$ of sufficiently large maximum degree $Δ$ satisfies \\[ q_B(G)\\le \\begin{cases} Δ+Δ^{1-η}+1, & \\text{if }s\\le2,\\\\ (4h-2)(Δ-1)+1, & \\text{if }s\\ge3\\text{ and }k\\le1,\\\\ C\\frac{Δ^{2-\\frac{1}{k}}}{\\logΔ}, & \\text{if }k\\ge2, \\end{cases} \\] where $η>0$ and $C>0$ depend only on $F$. For $k\\le1$, the linear order is best possible, and for $k\\ge2$, the bound is nearly sharp. To prove these results, we develop a common reduction of the coloring problems to $P$-perfect matching problems in auxiliary hypergraphs and apply the forbidden-submatching theorem of Delcourt and Postle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39716",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhijun Lu"
    },
    {
      "name": "Qirui Ying"
    },
    {
      "name": "Huimin Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1v55rtag3enty.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39716",
  "fulltext_url": null,
  "id": "1v55rtag3enty",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39716",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7215033588",
  "pdf_url": "https://arxiv.org/pdf/2609.39716",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1v55rtag3enty.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39716",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39716"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$(2,\\mathcal{F})$-Avoiding Coloring and B-Coloring under Bipartite Exclusions",
  "title_text": "(2,ℱ)-Avoiding Coloring and B-Coloring under Bipartite Exclusions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1v55rtag3enty",
  "venue": "arXiv",
  "versions": []
}