Linearity Property of Unique Colourings in Random Graphs

TL;DR

This paper investigates how the minimum number of colours needed for unique colourings in random graphs scales linearly with a specified uniqueness parameter, extending previous concepts like conflict-free and injective colourings.

Contribution

It introduces a new linearity property for unique colourings in random graphs and analyzes how vertex neighbourhoods affect this property using probabilistic methods.

Findings

Minimum colours grow linearly with the uniqueness parameter

Vertex neighbourhood size influences the linearity property

Example with tree unique colourings illustrates the concept

Abstract

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.