Geodesic
On weakly Turán-good graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1539-1550

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Given graphs H and F with χ(H) lt;χ(F), we say that H is weakly F-Turán-good if among n-vertex F-free graphs, a (χ(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_2ℓ+1-Turán-good for every k gt;ℓ.
Keywords: generalized Tur\'an problem, extremal, Tur\'an-good 
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     author = {Gerbner, D\'aniel},
     title = {On weakly {Tur\'an-good} graphs},
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     number = {4},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a15/}
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Gerbner, Dániel. On weakly Turán-good graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1539-1550. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a15/