Colourings, transversals and local sparsity

TL;DR

This paper proves a new result on graph colourings and independent transversals under local sparsity conditions, extending previous work and using the semi-random method.

Contribution

It introduces a generalized theorem linking local sparsity and transversals, strengthening prior results by Loh, Sudakov, Molloy, and Thron.

Findings

Establishes conditions for the existence of independent transversals in graphs.

Extends previous results on list colouring and local sparsity.

Uses semi-random method to prove the main theorem.

Abstract

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).