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arXiv 2102.00323math.CO
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长度为3的路径是$K_{r+1}$ -Turán Good

Paths of Length Three are $K_{r+1}$-Turán Good

Kyle Murphy, JD Nir

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24 pages
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摘要：

广义Turán问题$ext(n,T,F)$是确定图$T$在$n$顶点上可以存在于$F$ -free图中的最大拷贝数。最近，Gerbner和Palmer注意到广义Turán问题的解通常是原始的Turán图。他们的名字“$F$ -图兰好”的图形$T$，其中，对于足够大的$n$，解决方案的广义图兰问题是实现了图兰图。他们证明了两条边上的路径图$P_2$对所有$r \ge 3$都是$K_{r+1}$ -Turán-好的，但他们猜想同样的结果对所有$P_\ell$都应该成立。本文利用基于旗代数的论证，证明了三条边上的路$P_3$对所有$r \ge 3$也是$K_{r+1}$ -Turán-好的.

英文摘要：

The generalized Turán problem $ext(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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