The maximum number of pentagons in a planar graph

Abstract

Abstract In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an ‐vertex planar graph. They precisely determined the maximum number of triangles and four‐cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the ‐vertex, planar graphs with the maximum number of pentagons.