Colorings, transversals, and local sparsity

• Published in: Random structures & algorithms Vol. 61; no. 1; pp. 173 - 192
• Main Authors: Kang, Ross J., Kelly, Tom
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.08.2022; Wiley Subscription Services, Inc
• Subjects: Apexes; conflict choosability; correspondence coloring; Graph theory; independent transversals; list coloring; Sparsity
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.21051

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).