Colorings, transversals, and local sparsity. Issue 1 (13th October 2021) - Kang, Ross J.; Kelly, Tom
Abstract: 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 μ satisfying μ ( d ) = o ( d ) as d → ∞, there is a function λ satisfying λ ( d ) = d + o ( d ) as d → ∞ such that the following holds. For any graph H and any partition of its vertices into parts of size at least λ 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 μ, 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).