Geodesic
On the maximum number of odd cycles in graphs without smaller odd cycles
Andrzej Grzesik1 ; Bartłomiej Kielak1 ; Andrzej Grzesik ; Bartłomiej Kielak
1Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 755-758 
Andrzej Grzesik; Bartłomiej Kielak; Andrzej Grzesik; Bartłomiej Kielak. On the maximum number of odd cycles in graphs without smaller odd cycles. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 755-758. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a61/
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     title = { On the maximum number of odd cycles in graphs without smaller odd cycles},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {755--758},
     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a61/}
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Voir la notice de l'article provenant de la source Comenius University

We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length~$k$. This generalizes the previous results on the maximum number of pentagons in triangle-free graphs, conjectured by Erd\H os in 1984, and asymptotically determines the generalized Tur\'an number $\mathrm{ex}(n,C_k,C_{k-2})$ for odd $k$. In contrast to the previous results on the pentagon case, our proof is not computer-assisted.