  We study the occurrence of physical collisions between stars in young and
compact star cluster. The calculations are performed on the GRAPE-4 with the
starlab software environment which include the dynamical evolution and the
nuclear evolution of all stars and binaries. The selection of the initial
conditions is based on existing and well observed star clusters, such as R136
in the 30 Doradus region in the Large Magellanic Cloud and the Arches and
Quintuplet star clusters in the vicinity of the Galactic center.
  Collisions between stars occurred rather frequently in our models. At any
time a single star dominates the collision history of the system. The collision
rate of this runaway merger scales with the initial relaxation time of the
cluster and is independent on other cluster parameters, such as the initial
mass function or the initial density profile of the cluster. Subsequent
encounters result in a steady grow in mass of the coagulating star, until it
escapes or explodes in a supernova. The collision rate in these models is about
0.00022 collisions per star per Myr for a cluster with an initial relaxation
time of 1 Myr.
