The maximum number of paths of length three in a planar graph

Abstract

Let f ( n , H ) $f(n,H)$ denote the maximum number of copies of H $H$ possible in an n $n$-vertex planar graph. The function f ( n , H ) $f(n,H)$ has been determined when H $H$ is a cycle of length 3 or 4 by Hakimi and Schmeichel and when H $H$ is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine f ( n , H ) $f(n,H)$ exactly in the case when H $H$ is a path of length 3.