Improper colouring of (random) unit disk graphs

Abstract : For any graph $G$, the $k$-improper chromatic number $\chi^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to generalise results where only proper colouring was considered.