  We study a P\'olya-type urn model defined as follows. Start at time 0 with a
single ball of some colour. Then, at each time n>0, choose a ball from the urn
uniformly at random. With probability 1/2<p<1, return the ball to the urn along
with another ball of the same colour. With probability 1-p, recolour the ball
to a new colour and then return it to the urn. This is equivalent to the
supercritical case of a random graph model studied by Backhausz and M\'ori, and
Th\"ornblad. We prove that, with probability 1, there is a dominating colour,
in the sense that, after some random but finite time, there is a colour that
always has the most number of balls. A crucial part of the proof is the
analysis of an urn model with two colours, in which the observed ball is
returned to the urn along with another ball of the same colour with probability
p, and removed with probability 1-p. Our results here generalise a classical
result about the P\'olya urn model (which corresponds to p=1).
