Separation choosability and dense bipartite induced subgraphs

TL;DR

This paper introduces the concept of separation choosability in list coloring, demonstrating its growth with minimum degree in bipartite graphs and exploring related Ramsey-type questions for triangle-free graphs.

Contribution

It defines separation choosability, proves its relation to minimum degree in bipartite graphs, and raises new Ramsey-type questions for triangle-free graphs.

Findings

Separation choosability increases with the logarithm of minimum degree in bipartite graphs.

Strengthens previous bounds by Molloy, Thron, and Alon.

Poses new questions on bipartite induced subgraphs in triangle-free graphs.

Abstract

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree $d$ contain a bipartite induced subgraph of minimum degree $\Omega(\log d)$ as $d\to\infty$?