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 $r \ge 3$.
@article{10_37236_10225,
author = {Kyle Murphy and JD Nir},
title = {Paths of length three are {\(K_{r+1}\)-Tur\'an-good}},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10225},
zbl = {1486.05148},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10225/}
}
TY - JOUR
AU - Kyle Murphy
AU - JD Nir
TI - Paths of length three are \(K_{r+1}\)-Turán-good
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10225/
DO - 10.37236/10225
ID - 10_37236_10225
ER -
%0 Journal Article
%A Kyle Murphy
%A JD Nir
%T Paths of length three are \(K_{r+1}\)-Turán-good
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10225/
%R 10.37236/10225
%F 10_37236_10225
Kyle Murphy; JD Nir. Paths of length three are \(K_{r+1}\)-Turán-good. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10225
