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Title:
On weakly Turán-good graphs
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Discussiones Mathematicae Graph Theory 44(4) (2024) 1539-1550
Received: 2022-12-27 , Revised: 2023-07-04 , Accepted: 2023-07-07 , Available online: 2023-08-22 , https://doi.org/10.7151/dmgt.2510
Abstract:
Given graphs $H$ and $F$ with $\chi(H)<\chi(F)$, we say that $H$ is weakly
$F$-Turán-good if among $n$-vertex $F$-free graphs, a $(\chi(F)-1)$-partite
graph contains the most copies of $H$. Let $H$ be a bipartite graph that
contains a complete bipartite subgraph $K$ such that each vertex of $H$ is
adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Turán-good,
improving a very recent asymptotic bound due to Grzesik, Győri, Salia and
Tompkins. They also showed that for any $r$ there exist graphs that are not
weakly $K_r$-Turán-good. We show that for any non-bipartite $F$ there exist
graphs that are not weakly $F$-Turán-good. We also show
examples of graphs that are $C_{2k+1}$-Turán-good but not
$C_{2\ell+1}$-Turán-good for every $k>\ell$.
Keywords:
generalized Turán problem, extremal, Turán-good
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