The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps
Abstract
For a finite abelian group Γ, a map f: V(G)→Γ 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 1 for the all-ones vector indexed by V(G), and let A̅ denote the endomorphism induced by A on Λ1=ℤV(G)/ℤ1. When A̅ is nonsingular over ℚ, define the reduced Smith group by Sred(G)=cokerA̅. In previous work, we established that for every regular graph G of positive degree with A̅ nonsingular over ℚ, one has G admits an affinely generating Γ−magic map Longleftrightarrow Γ↪Sred(G). We determine this group explicitly for Kneser graphs. If r≥1, n≥2r, and mj=binom nj−binom nj−1, then Sred(K(n,r))≅ ⨁j=1r (ℤ/binomn−r−jr−jℤ)mj. 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≅ℤ/6ℤ⊕ℤ/6ℤ. 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≥1 and n≥2r.
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BibTeX
@misc{batal2026reduced,
title = {{The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps}},
author = {Ahmet Batal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12546},
eprint = {2609.12546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12546},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Reduced Smith Group of a Kneser Graph and Its Application to Group-Valued Magic Maps
AU - Ahmet Batal
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12546
UR - https://arxiv.org/abs/2609.12546
AB - For a finite abelian group Γ, a map f: V(G)→Γ 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 1 for the all-ones vector indexed by V(G), and let A̅ denote the endomorphism induced by A on Λ_1=ℤ^V(G)/ℤ1. When A̅ is nonsingular over ℚ, define the reduced Smith group by S_red(G)=cokerA̅. In previous work, we established that for every regular graph G of positive degree with A̅ nonsingular over ℚ, one has G admits an affinely generating Γ−magic map Longleftrightarrow Γ↪S_red(G). We determine this group explicitly for Kneser graphs. If r≥1, n≥2r, and m_j=binom nj−binom nj−1, then S_red(K(n,r))≅ ⨁_j=1^r (ℤ/binomn−r−jr−jℤ)^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≅ℤ/6ℤ⊕ℤ/6ℤ. 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≥1 and n≥2r.
ER -