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

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 $r \ge 3$.

Some exact results of the generalized Turán numbers for paths

“…For Conjecture 1.2, Murphy and Nir [16], and Qian et al [18]) have confirmed this conjecture for P 4 , P 5 and k ≥ 4, we continue to confirm this conjecture for P 6 and k ≥ 4 by showing the following a little more generalized result. The rest of this paper is organized as follows.…”

“…We leave this as an open problem. It has been shown that P 2 (Simonovits [20]), P 3 (Gerbner and Palmer [11]), P 4 (Murphy and Nir [16]), P 5 (Qian et al [18]), and P 6 (Theorem 1.5) have the weak T-property. some j = i, 3 ≤ j ≤ k − 1.…”

“…Clearly, P ℓ is a connected bipartite graph with ν(P ℓ ) = ⌊ ℓ 2 ⌋. The (ii) (Simonovits [20]), (viii) (Gerbner and Palmer [11]) and (ix) (Murphy and Nir [16], Qian et al [18]) imply that P ℓ has the weak T-property when ℓ ≤ 5. To get the result of Theorem 1.5 (a), it is sufficient to show that P 6 has the weak T-property.…”

Some exact results for generalized Turán problems

“…This approach where we prove that H is F -Turán-stable and use it to prove that H is F -Turán-good can also be found in [18,21,17,11]. Finally, Gerbner [13] provided the following general formulation.…”

Paths are Turán-good

“…Results of Gerbner [1] imply that each path is F -Turán-good for a large class of 3-chromatic graphs with color-critical edges, including odd cycles. Murphy and Nir [7] showed that P 4 is K k+1 -Turán-good. Qian, Xie and Ge [8] showed that P 5 is K k+1 -Turán-good.…”