{
  "abstract": "We study the minimum coefficient degree $N_{k,\\F}(G)$ of a Nullstellensatz certificate for Bayer's $k$-coloring equations, where the characteristic of $\\F$ does not divide $k$. If $J$ is a \\HJ\\ join of non-$k$-colorable graphs $G,H$ and $m=\\max\\{N_{k,\\F}(G),N_{k,\\F}(H)\\}$, then $N_{k,\\F}(J)\\leq m+k$. When deletion of the selected edge makes each input $k$-colorable, we also have $N_{k,\\F}(J)\\geq m$; the degree congruence then gives $N_{k,\\F}(J)\\in\\{m,m+k\\}$. This partially answers a question of Li, Lowenstein, and Omar. For three-coloring over $\\F_2$, we construct an infinite $4$-critical family of exact degree seven, attaining the bound at input degree four. In contrast, every graph constructed from $K_4$ solely by \\HJ\\ joins has degree $O(\\log n)$ and a certificate with polynomially many terms: joins preserve treewidth at most three, and balanced separators yield low-degree certificates. Additional vertex identifications are excluded from this obstruction. We classify all single identifications of the $25$-vertex base graph; exactly $36$ preserve degree seven, producing $24$-vertex $4$-critical graphs of treewidth four. A compressed self-join at adjacent true twins prevents degree loss and gives a repeatable rule adding four vertices per round. The rule does not establish degree amplification or preservation of criticality. Exact witnesses and standalone verification programs accompany the finite results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14865",
  "author_total": 1,
  "authors": [
    {
      "name": "Ying Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5jzbzv9x6xatc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14865",
  "fulltext_url": null,
  "id": "5jzbzv9x6xatc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14865",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213310583",
  "pdf_url": "https://arxiv.org/pdf/2609.14865",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5jzbzv9x6xatc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14865",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14865"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nullstellensatz degree under Hajós joins and vertex identifications",
  "title_text": "Nullstellensatz degree under Hajós joins and vertex identifications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/5jzbzv9x6xatc",
  "venue": "arXiv",
  "versions": []
}