arXiv:2110.06921v1 [math.CO] 13 Oct 2021 Generalized outerplanar Tur´ an number of short paths Ervin Gy˝ ori 1,2 Addisu Paulos 2 Chuanqi Xiao 2 1 Alfr´ ed R´ enyi Institute of Mathematics, Budapest gyori.ervin@renyi.hu 2 Central European University, Budapest addisu_2004@yahoo.com, chuanqixm@gmail.com Abstract Let H be a graph. The generalized outerplanar Tur´ an number of H , denoted by f OP (n, H ), is the maximum number of copies of H in an n-vertex outerplanar graph. Let P k be the path on k vertices. In this paper we give an exact value of f OP (n, P 4 ) and a best asymptotic value of f OP (n, P 5 ). Moreover, we characterize all outerplanar graphs containing f OP (n, P 4 ) copies of P 4 . 1 Introduction In 1941, Tur´ an [13] proved a classical result in the field of extremal graph theory. He determined exactly the maximum number of edges an n-vertex K r -free graph may contains. After his result, for a graph H , the maximum number of edges in an n-vertex H -free graph, denoted by ex(n, H ), is named as Tur´ an number of H . A major breakthrough in the study of the Tur´ an number of graphs came in 1966, with the proof of the famous theorem by Erd˝ os, Stone and Simonovits [6, 7]. They determined an asymptotic value of the Tur´ an number of any non-bipartite graph H . In particular, they proved ex(n, H )=  1 − 1 χ(H)−1  ( n 2 ) + o(n 2 ), where χ(H ) is the chromatic number of H . Since then researchers have been interested working on Tur´ an number of class of bipartite (degenerate) graphs and extremal graph problems with some more generality. Determining the maximum number of copies of H in an n-vertex F -free graph, denoted by ex(n, H, F ), is among such problems. Since we count the number of copies of a given graph which is not necessarily an edge, such an extremal graph problem is commonly named as generalized Tur´ an problem. The results on ex(n, K r ,K t ) by Zykov [14] (and independently by Erd˝ os [5]), ex(n, C 5 ,C 3 ) by Gy˝ ori [10] and ex(n, C 3 ,C 5 ) by Bollob´ as and Gy˝ ori [9] were sporadic initial contributions. 1