图到阿贝尔Cayley图的诱导嵌入
Induced Embeddings of Graphs into Abelian Cayley Graphs
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中文总结 AI 辅助
该研究将图的阿贝尔Cayley图诱导嵌入最小群阶作为图不变量，给出局部下界，确定了路径、完全二部图等多类图的精确值，分析了循环群限制的代价并找到相关极值图。
中文摘要 AI 辅助
对于具有$n$个顶点的有限图$G$，设$η(G)$表示满足$G$是$Γ$的某个Cayley图的诱导子图的有限阿贝尔群$Γ$的最小阶数。Babai和Sós（1985）确定了其最坏情况下的数量级为$Θ(n^2)$。我们转而将$η$视为单个图的不变量，在所有有限阿贝尔群上取最小值，而非仅在循环群上取最小值——后者是模$n$表示数相关文献中隐含的限制。我们证明了一个局部阶下界：$η(G)$至少为$n$与$G$的邻域的最大独立数的两倍两者中的最大值。这将Babai和Sós提出的诱导星型图与无和集之间的对应关系局部化到任意顶点；阿贝尔群中最大无和集分类的一个推论不会降低该下界，但会限制哪些宿主阶是可容许的，从而缩减搜索范围。我们精确确定了路径图的$η$值为$n+1$，完全二部图$K_{a,b}$的$η$值为$2\max(a,b)$且达到该下界。笛卡尔积界给出$η(P_m \\,\square\\, P_m) = (1+o(1))n$。我们报告了22个图的经认证的$η$精确值，这些值是在所有阿贝尔群上计算得到的。22个最优宿主中有17个是循环群，因此在大多数这些图上，循环群限制没有代价；但在该限制起作用的情况下，代价很高。仅针对循环群的搜索得到Petersen图的结果为36，而真实值为16；Frucht图的结果为59，而真实值为27。该限制的代价是集中而非分散的，我们确定了需要付出该代价的图。我们还精确确定了满足$2 \le q \le 6$的双星图$D_{q,q}$的$η$值，每种情况均为$5q$。由于$η(D_{6,6}) = 30$超过$2n = 28$，因此不存在低于$15/7$的常数能够对所有树的$η(T)/n$进行上界约束。
英文摘要
For a finite graph $G$ on $n$ vertices, let $η(G)$ denote the least order of a finite abelian group $Γ$ for which $G$ is an induced subgraph of some Cayley graph of $Γ$. Babai and Sós (1985) settled the worst-case order of magnitude: it is $Θ(n^2)$. We treat $η$ instead as an invariant of the individual graph, minimised over all finite abelian groups rather than over the cyclic groups alone, which is the restriction implicit in the literature on representation numbers modulo $n$. We prove a local order floor: $η(G)$ is at least the maximum of $n$ and twice the largest independence number of a neighbourhood of $G$. This localises at an arbitrary vertex the correspondence of Babai and Sós between induced stars and sum-free sets; a corollary of the classification of maximum sum-free sets in abelian groups does not lower this floor, but restricts which host orders are admissible and so prunes the search. We determine $η$ exactly for paths, where it equals $n+1$, and for complete bipartite graphs $K_{a,b}$, where it equals $2\max(a,b)$ and meets the floor. A Cartesian product bound gives $η(P_m \,\square\, P_m) = (1+o(1))n$. We report certified exact values of $η$ for $22$ graphs, computed over all abelian groups. Seventeen of the $22$ optimal hosts are cyclic, so on most of these graphs the cyclic restriction costs nothing; where it bites, however, it is expensive. A search restricted to cyclic groups returns $36$ for the Petersen graph against the true value $16$, and $59$ for the Frucht graph against $27$. The cost of the restriction is concentrated rather than diffuse, and we identify the graphs on which it is paid. We also determine $η$ exactly for the double stars $D_{q,q}$ with $2 \le q \le 6$, obtaining $5q$ in each case. Since $η(D_{6,6}) = 30$ exceeds $2n = 28$, no constant below $15/7$ can bound $η(T)/n$ over all trees.
