Extremal C4-free/C5-free planar graphs

Abstract

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,C4) ≤ 157(n-2) for all n ≥ 4 and ex_P(n,C5) ≤ 12n-335 for all n ≥ 11, and showing that both of these bounds are tight.

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