Linearity Property of Unique Colourings in Random Graphs

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.