Colourings, transversals and local sparsity

Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $\mu$ satisfying $\mu(d)=o(d)$ as $d\to\infty$, there is a function $\lambda$ satisfying $\lambda(d)=d+o(d)$ as $d\to\infty$ such that the following holds. For any graph $H$ and any partition of its vertices into parts of size at least $\lambda$ such that (a) for each part the average over its vertices of degree to other parts is at most $d$, and (b) the maximum degree from a vertex to some other part is at most $\mu$, there is guaranteed to be a transversal of the parts that forms an independent set of $H$. This is a common strengthening of two results of Loh and Sudakov (2007) and Molloy and Thron (2012), each of which in turn implies an earlier result of Reed and Sudakov (2002).