The Maximum Number of Pentagons in a Planar Graph

Abstract

Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an n-vertex planar graph. They determined this number exactly for triangles and 4cycles and conjectured the solution to the problem for 5-cycles. We confirm their conjecture.

Planar graphs with the maximum number of induced 4-cycles or 5-cycles

“…Also, it is straightforward to see that the complete bipartite graph K 2,n−2 contains 1 2 (n 2 − 5n + 6) induced 4-cycles, so f I (n, C 4 ) = 1 2 n 2 + O(n), as observed in [7]. In [11] Győri, Paulos, Salia, Tompkins, and Zamora determined f (n, C 5 ) exactly for all n ≥ 5, and in [7] Ghosh, Győri, Janzer, Paulos, Salia, and Zamora showed that f I (n, C 5 ) = 1 3 n 2 + O(n). In [3] and [4] Cox and Martin determined f (n, H) asymptotically when H is a small even cycle, showing that f (n, C 2k ) = n k k + o(n k ) for k ∈ {3, 4, 5, 6}.…”

“…, R t , where R i has boundary uv i wv i+1 , with the addition in the subscript taken modulo t. If at least 7 n 4/5 of these regions have vertices in their interior, then the drawing of G contains a K 2,7 n 4/5 subgraph which does not contain an empty K 2,7 . But G also does not contain a vertex in at most 11 20 n induced 5-cycles, contradicting Lemma 13. So fewer than 7 n 4/5 of the regions have any vertices in their interior. Thus for all but o(n) values of i, none of the regions R i−3 , R i−2 , .…”

“…Let n be large and let G be an n-vertex plane graph. Then either G contains an empty K 2,7 or some vertex of G is in at most 11 20 n induced 5-cycles.…”

“…Let n be large and let G be an n-vertex plane graph. If G contains a K 2,7 n 4/5 subgraph, then either this K 2,7 n 4/5 contains an empty K 2,7 or G contains a vertex in at most 11 20 n induced 5-cycles.…”

“…Hakimi and Schmeichel showed in [12] that f (n, C 2k+1 ) = O(n k ), and this seems to be the best known upper bound. For results on f (n, H) for various other graphs H see [1], [3], [4], [5], [9], [10], [11], [18], and [19].…”