Extremal C4‐Free/C5‐Free Planar Graphs

We study the topic of "extremal" planar graphs, defining ex(P) (n, H) to be the maximum number of edges possible in a planar graph on n vertices that does not contain a given graph H as a subgraph. In particular, we examine the case when H is a small cycle, obtaining ex(P) (n, C-4) <= 15/7 (n-2) for all n >= 4 and ex(P) (n, C-5) = 12n-33/5 for all n >= 11, and showing that both of these bounds are tight. (C) 2015 Wiley Periodicals, Inc.