  In this paper, we study unique colourings in random graphs as a
generalization of both conflict-free and injective colourings. Specifically, we
impose the condition that a fraction of vertices in the neighbourhood of any
vertex are assigned unique colours and use vertex partitioning and the
probabilistic method to show that the minimum number of colours needed grows
linearly with the uniqueness parameter, unlike both conflict-free and injective
colourings. We argue how the unboundedness of the vertex neighbourhoods
influences the linearity property and illustrate our case with an example
involving treeunique colourings in random graphs.
