The Maximum Number of Paths of Length Three in a Planar Graph

Abstract

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

The Maximum Number of Pentagons in a Planar Graph

“…The exact value of f (n, K 4 ) was determined by Wood in [14]. In [8], we determined f (n, P 3 ), where P 3 is the path with 3 edges.…”

The maximum number of 10- and 12-cycles in a planar graph

“…Alon and Caro [1] continued this study by determining N P (n, K 2,k ) exactly for all k; in particular, they determined N P (n, P 3 ). Győri et al [4] later gave the exact value for N P (n, P 4 ), and the same authors determined N P (n, C 5 ) in [5]. Afterward, Ghosh et al [3] asymptotically determined N P (n, P 5 ), and, very recently, the current authors [2] computed N P (n, P 7 ) asymptotically.…”

“…In [2] a general technique was introduced which allows one to bound N P (n, H) whenever H exhibits a particular subdivision structure. Using this technique, the authors established N P (n, C 6 ) ∼ (n/3) 3 and N P (n, C 8 ) ∼ (n/4) 4 . They furthermore conjectured that…”

Counting paths, cycles and blow-ups in planar graphs

“…Alon and Caro [1] continued this study by determining N P (n, K 2,k ) exactly for all k; in particular, they determined N P (n, P 3 ). Győri et al [4] later gave the exact value for N P (n, P 4 ), and the same authors determined N P (n, C 5 ) in [5].…”