含大\(F -\)稀疏集的图的图兰型界
Turán-Type Bounds for Graphs Containing Large $F$-Sparse Sets
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中文总结 AI 辅助
研究含大\(F -\)稀疏集的图的图兰型极值问题，证明了特定条件下\(K_{r + 1}\) - 自由图边数的不等式，刻画了等式情况，还给出了禁止图及一般\(H -\)自由图的相关扩展。
中文摘要 AI 辅助
我们研究了含大\(F -\)稀疏顶点集的图的图兰型极值问题，即其诱导子图包含少量\(F\)副本的顶点集。对于整数\(r > s\geq1\)，我们证明若\(n\)个顶点的\(K_{r + 1}\) - 自由图\(G\)包含大小为\(m\geq\lceil sn/r\rceil\)的集合\(M\)，使得\(G[M]\)是\(K_{s + 1}\) - 自由的，则\(e(G)\leq m(n - m)+t_s(m)+t_{r - s}(n - m)\)。我们刻画了等式成立的情况为完全\(r -\)部图，其顶点类分为两个平衡组，总大小分别为\(m\)和\(n - m\)，分别由\(s\)个和\(r - s\)个类组成。我们还证明了嵌入到两个边关键图的并中的禁止图及一般\(H -\)自由图的色关键扩展和渐近扩展，其中规定的大顶点集跨越少量固定图\(F\)的副本且\(\chi(F)<\chi(H)\)。
英文摘要
We study Turán-type extremal problems for graphs containing a large $F$-sparse vertex set, meaning a vertex set whose induced subgraph contains few copies of $F$. For integers $r>s\ge 1$, we prove that if a $K_{r+1}$-free graph $G$ on $n$ vertices contains a set $M$ of size $m\ge \lceil sn/r\rceil$ such that $G[M]$ is $K_{s+1}$-free, then \[ e(G)\le m(n-m)+t_s(m)+t_{r-s}(n-m). \] We characterize the equality cases as the complete $r$-partite graphs whose vertex classes split into two balanced groups of total sizes $m$ and $n-m$, consisting of $s$ and $r-s$ classes, respectively. We also prove a color-critical extension for forbidden graphs that embed into a join of two edge-critical graphs, together with an asymptotic extension for general $H$-free graphs in which the prescribed large vertex set spans few copies of a fixed graph $F$ with $χ(F)<χ(H)$.
