Speaker: Adrian Dumitrescu
Title: On peeling sequences and edge partitions of complete geometric graph
Abstract: (I) Given a set of $n$ labeled points in general position in the plane,
we remove all of its points one by one. At each step, one point from
the convex hull of the remaining set is erased.
In how many ways can the process be carried out?
The answer obviously depends on the point set.
If the points are in convex position, there are exactly $n!$ ways,
which is the maximum number of ways for $n$ points. But what is the minimum number?
It is shown that this number is (roughly) at least $3^n$ and at most $12.29^n$.
This is joint work with Geza Toth
(II) A \emph{complete geometric graph} consists of a set $P$ of $n$ points
in the plane, in general position, and all segments (edges) connecting
them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood,
whether there exists a positive constant $c<1$, such that every
complete geometric graph on $n$ points can be partitioned into at most
$cn$ plane graphs (that is, noncrossing subgraphs). We answer this
question in the affirmative in the special case where the underlying
point set $P$ is \emph{dense}, which means that the ratio between the
maximum and the minimum distances in $P$ is of the order of $\Theta(\sqrt{n})$.
This is joint work with Janos Pach