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arXiv · 2102.00323

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

Kyle Murphy·JD Nir
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Abstract

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 $r \ge 3$.

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Kyle Murphy, JD Nir. 2021-01-30. Paths of Length Three are $K_{r+1}$-Turán Good. https://doi.org/10.37236/10225

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