Colorings, transversals, and local sparsity

Ross J. Kang, Tom Kelly

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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 urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0001 satisfying urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0002 as urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0003, there is a function urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0004 satisfying urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0005 as urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0006 such that the following holds. For any graph H and any partition of its vertices into parts of size at least urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0007 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 urn:x-wiley:rsa:media:rsa21051:rsa21051-math-0008, 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).
Original languageEnglish
Number of pages20
JournalRandom Structures and Algorithms
Early online date13 Oct 2021
DOIs
Publication statusE-pub ahead of print - 13 Oct 2021

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