Abstract
The generalized Turán problem ex$(n,T,F)$ 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 $r \ge 3$.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
4 articles.
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1. Some Exact Results for Non-Degenerate Generalized Turán Problems;The Electronic Journal of Combinatorics;2023-12-15
2. The Cycle of Length Four is Strictly F-Turán-Good;Bulletin of the Malaysian Mathematical Sciences Society;2023-11-08
3. Some exact results of the generalized Turán numbers for paths;European Journal of Combinatorics;2023-05
4. Paths are Turán-good;Graphs and Combinatorics;2023-05