Outerplanar Turán numbers of cycles and paths

Abstract

A graph is outerplanar if it has a planar embedding such that all its vertices lie on the outer face. Given a graph H, the planar (resp. outerplanar) Turán number of H, denoted by exP(n,H) (resp. exOP(n,H)), is the maximum number of edges over all planar (resp. outerplanar) graphs on n vertices which do not contain a copy of H. Dowden (2016) [3] initiated the study of planar Turán numbers of cycles. Recently, Lan, Shi and Song investigated the planar Turán numbers of paths and some other important graphs. Up to now, both exP(n,Ck) and exP(n,Pk) are open for general k, even a sharp upper bound of which is still unknown. In this paper, the outerplanar Turán numbers of cycles and paths are completely determined. A key approach is to establish an equivalence relation, i.e., k-face-connectedness, by which we can decompose an outerplanar graph into some so-called trivial k-blocks. This presents a probable way to consider planar Turán numbers or outerplanar Turán numbers of graphs.