  The Hadwiger-Nelson problem asks for the minimum number of colors, so that
each point of the plane can be assigned a single color with the property that
no two points unit-distance apart are identically colored. It is now known that
the answer is $5$, $6$, or $7$, Here we consider the problem in the context of
Minkowski planes, where the unit circle is a regular polygon with $8$, $10$, or
$12$ vertices. We prove that in each of these cases, one also needs at least
five colors.
