论文标题
以$ k_r $ -free Graphs的形式最大化五速
Maximizing five-cycles in $K_r$-free graphs
论文作者
论文摘要
Erdős五角大楼问题要求找到一个$ n $ vertex的无三角形图，该图可最大化$ 5 $ -CYCLE的数量。 Grzesik使用FLAG代数解决了问题，并由Hatami，Hladký，Král'，Norin和Razborov独立解决。最近，帕尔默（Palmer）提出了最大化$ 5 $ - 周期为$ k_ {k+1} $ - 免费图形的总体问题。使用标志代数，我们表明每$ k_ {k + 1} $ - 免费的订单$ n $图形最多包含\ [\ frac {1} {10k^4}（k^4-5k^3 + 10k^3 + 10k^2-10k + 4）足够大的$ n $。
The Erdős Pentagon problem asks to find an $n$-vertex triangle-free graph that is maximizing the number of $5$-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladký, Král', Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of $5$-cycles in $K_{k+1}$-free graphs. Using flag algebras, we show that every $K_{k+1}$-free graph of order $n$ contains at most \[\frac{1}{10k^4}(k^4 - 5k^3 + 10k^2 - 10k + 4)n^5 + o(n^5)\] copies of $C_5$ for any $k \geq 3$, with the Turán graph begin the extremal graph for large enough $n$.
