Finding the maximum number of induced cycles of length k in a graph on n vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidický and Pfender answered the question in the case k = 5. In this paper we show that an n-vertex planar graph contains at most n 2 3 + O(n) induced C 5 's, which is asymptotically tight.
Let ex P (n, T, H) denote the maximum number of copies of T in an n-vertex planar graph which does not contain H as a subgraph. When T = K 2 , ex P (n, T, H) is the well studied function, the planar Turán number of H, denoted by ex P (n, H). The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ex P (n, C 4 ) and ex P (n, C 5 ). Later on, Y. Lan, et al. continued this topic and proved that ex P (n, C 6 ) ≤ 18(n−2)
7. In this paper, we give a sharp upper bound ex P (n, C 6 ) ≤ 5 2 n − 7, for all n ≥ 18, which improves Lan's result. We also pose a conjecture on ex P (n, C k ), for k ≥ 7.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.