Nika Salia: Extremal problems in planar graphs

Abstract:

This will be a very special survey type  talk, where authors
will attempt to popularise a set of old forgotten problems.  We will
survey our fresh results and give quite a few delicate problems for
further investigation. Our planar project was inspired by an old result
of  Hakimi and Schmeichel. In 1979, they considered the problem of
maximizing the number of cycles of a given length in an $n$-vertex
planar graph.  They precisely determined the maximum number of triangles
and $4$-cycles and presented a conjecture for the maximum number of
pentagons. We confirmed their conjecture and characterized the
$n$-vertex, planar graphs with the maximum number of pentagons. Since
then we got a number of different results about induced pentagons, and
psaths of length 3 and 4: arXiv: 1909.13532, 1909.13539, 2002.04579,