AI 中文总结
本文确定了 Kneser 图 K(n,r) 的约化 Smith 群，并利用嵌入准则证明：除 (n,r)=(9,2) 且 Γ≅Z/6Z⊕Z/6Z 外，K(n,r) 不存在仿射生成 Γ-幻映射，且整个范围内无 Γ-距离幻标号。
AI 中文摘要
对于有限阿贝尔群 $\Gamma$，若映射 $f\colon V(G)\to\Gamma$ 满足其所有邻点值之和与顶点无关，则称 $f$ 为 $\Gamma$-幻映射。若其两两差值生成 $\Gamma$，则称其为仿射生成映射。当 $|\Gamma|=|V(G)|$ 时，双射的 $\Gamma$-幻映射即为 $\Gamma$-距离幻标号。对于具有邻接矩阵 $A$ 的正则图 $G$，记 $\mathbf1$ 为以 $V(G)$ 为索引的全一向量，并令 $\overline A$ 表示 $A$ 在 $\Lambda_{\mathbf1}=\mathbb Z^{V(G)}/\mathbb Z\mathbf1$ 上诱导的自同态。当 $\overline A$ 在 $\mathbb Q$ 上非奇异时，定义约化 Smith 群为 $\mathsf S_{\mathrm{red}}(G)=\operatorname{coker}\overline A$。在先前的工作中，我们证明了对于每个具有正度且 $\overline A$ 在 $\mathbb Q$ 上非奇异的正则图 $G$，有 \\[ G\text{ 承认仿射生成 }\Gamma\text{-幻映射} \quad\Longleftrightarrow\quad \Gamma\hookrightarrow\mathsf S_{\mathrm{red}}(G). \\] 我们显式地确定了 Kneser 图的该群。若 $r\ge1$，$n\ge2r$，且 $m_j=\binom nj-\binom n{j-1}$，则 \\[ \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}. \\] 当标号群与 Kneser 图具有相同阶数时，所得嵌入准则仅剩一种情形存在仿射生成映射。更精确地，若 $\Gamma$ 是阶为 $\binom nr$ 的阿贝尔群，则 $K(n,r)$ 承认仿射生成 $\Gamma$-幻映射当且仅当 $(n,r)=(9,2)$ 且 $\Gamma\cong\mathbb Z/6\mathbb Z\oplus\mathbb Z/6\mathbb Z$。一个弱 Sidon 集界表明例外情形中的仿射生成映射不可能是双射。因此，在整个范围 $r\ge1$ 且 $n\ge2r$ 内，$K(n,r)$ 不承认 $\Gamma$-距离幻标号。
英文摘要
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 $n\ge2r$.
