  The Hadwiger number $H(J)$ of a topological disk $J$ in $\Re^2$ is the
maximal number of pairwise nonoverlapping translates of $J$ that touch $J$. It
is well known that for a convex disk, this number is six or eight. A conjecture
of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike
disk is at most eight. A. Bezdek proved that this number is at most seventy
five for any starlike disk. In this note, we prove that the Hadwiger number of
a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger
number of a topological disk $J$ such that $(\conv J) \setminus J$ is
connected, is six or eight.
