Colorings, transversals, and local sparsity

Motivated both by recently introduced forms of list coloring 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 ->infinity, there is a function lambda satisfying lambda(d)=d+o(d) as d ->infinity 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).