{
  "abstract": "For a finite abelian group $Γ$, a map $f\\colon V(G)\\toΓ$ is a $Γ$-magic map if the sum of its values over the neighbors of a vertex is independent of the vertex. It is affinely generating if its pairwise differences generate $Γ$. When $|Γ|=|V(G)|$, a bijective $Γ$-magic map is a $Γ$-distance magic labeling. For a regular graph $G$ with adjacency matrix $A$, write $\\mathbf1$ for the all-ones vector indexed by $V(G)$, and let $\\overline A$ denote the endomorphism induced by $A$ on $Λ_{\\mathbf1}=\\mathbb Z^{V(G)}/\\mathbb Z\\mathbf1$. When $\\overline A$ is nonsingular over $\\mathbb Q$, define the reduced Smith group by $\\mathsf S_{\\mathrm{red}}(G)=\\operatorname{coker}\\overline A$. In previous work, we established that for every regular graph $G$ of positive degree with $\\overline A$ nonsingular over $\\mathbb Q$, one has \\[ G\\text{ admits an affinely generating }Γ\\text{-magic map} \\quad\\Longleftrightarrow\\quad Γ\\hookrightarrow\\mathsf S_{\\mathrm{red}}(G). \\] We determine this group explicitly for Kneser graphs. If $r\\ge1$, $n\\ge2r$, and $m_j=\\binom nj-\\binom n{j-1}$, then \\[ \\mathsf S_{\\mathrm{red}}(K(n,r))\\cong \\bigoplus_{j=1}^{r} \\left(\\mathbb Z/\\binom{n-r-j}{r-j}\\mathbb Z\\right)^{m_j}. \\] When the labeling group and the Kneser graph have the same order, the resulting embedding criterion leaves only one case in which an affinely generating map exists. More precisely, if $Γ$ is an abelian group of order $\\binom nr$, then $K(n,r)$ admits an affinely generating $Γ$-magic map if and only if $(n,r)=(9,2)$ and $Γ\\cong\\mathbb Z/6\\mathbb Z\\oplus\\mathbb Z/6\\mathbb Z$. A weak-Sidon-set bound shows that the affinely generating maps in the exceptional case cannot be bijective. Hence $K(n,r)$ admits no $Γ$-distance magic labeling throughout the range $r\\ge1$ and $n\\ge2r$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12546",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmet Batal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b2997rg5q37tm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12546",
  "fulltext_url": null,
  "id": "b2997rg5q37tm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12546",
  "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": "W7212673233",
  "pdf_url": "https://arxiv.org/pdf/2609.12546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b2997rg5q37tm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps",
  "title_text": "The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/b2997rg5q37tm",
  "venue": "arXiv",
  "versions": []
}