在具有规定边数的图中最大化固定图的副本数
Maximizing copies of a fixed graph in graphs with a prescribed number of edges
浏览论文内容
中文总结 AI 辅助
研究在规定边数的图中最大化固定图副本数的问题，通过相关结果确定了特定图\(H\)的\(\operatorname{emb}(H,{\binom{n}{2}})\)值，还证明除匹配图外，\(K_n\)是使嵌入数最大的唯一图，刻画了\(K_n\)达最大嵌入数的图。
中文摘要 AI 辅助
对于图\(H\)，用\(\operatorname{emb}(H,m)\)表示\(H\)在大小为\(m\)的图中的最大标记嵌入数。Erdős提出了针对不同的\(H\)和\(m\)寻找\(\operatorname{emb}(H,m)\)的问题。基于相关渐近和稳定性结果，我们确定了对于每个分数独立数为\(v_H/2\)的图\(H\)以及所有足够大的\(n\)，\(\operatorname{emb}(H,{\binom{n}{2}})\)的值。我们还表明，对于那些\(H\)（匹配图除外）和\(n\)，具有最大嵌入数的唯一图是\(K_n\)。这完全刻画了\(K_n\)实现最大嵌入数的图。
英文摘要
For a graph $H$ denote by $\operatorname{emb}\left(H, m\right)$ the maximal number of labeled embeddings of $H$ in a graph of size $m$. Erdős posed the question of finding $\operatorname{emb}\left(H, m\right)$ for different values of $H$ and $m$. Following related asymptotic and stability results, we determine the value of $\operatorname{emb}\left(H, {\binom{n}{2}}\right)$ for every graph $H$ with fractional independence number $v_H/2$ and all sufficiently large $n$. We also show that for those $H$ (except for matchings) and $n$ the only graph with a maximal number of embeddings is $K_n$. This fully characterizes the graphs for which $K_n$ achieves the maximal number of embeddings.
