List colouring with a bounded palette

TL;DR

This paper investigates the bounds on list colouring with a bounded palette, establishing new lower bounds and improving upper bounds for the number of colours needed, depending on palette size and list size.

Contribution

It provides the first super-polynomial lower bounds on the palette size needed for list colouring, and improves existing upper bounds using container methods.

Findings

Lower bound on $C(k,2k-1)$ is $oldsymbol{ extstyle rac{4^k}{ oot{k} olinebreak}}$

Super-polynomial lower bounds on $C(k,oldsymbol{ extstyle o(k^2/ olinebreak ext{ln} k)})$

Improved upper bounds for $oldsymbol{ extstyle ext{if } olinebreak ext{l} olinebreak ext{ge } 2.75k}$

Abstract

Kr\'al' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists must be of size at least $k$. They showed that, for any integer $k\ge 2$, there is an integer $C=C(k,2k-1)$, satisfying $C = O(16^{k}\ln k)$ as $k\to \infty$, such that, if a graph is $(k,2k-1)$-choosable, then it is $C$-choosable, and asked if $C$ is required to be exponential in $k$. We demonstrate it must satisfy $C = \Omega(4^k/\sqrt{k})$. For an integer $\ell \ge 2k-1$, if $C(k,\ell)$ is the least integer such that a graph is $C(k,\ell)$-choosable if it is $(k,\ell)$-choosable, then we more generally supply a lower bound on $C(k,\ell)$, one that is super-polynomial in $k$ if $\ell = o(k^2/\ln k)$, by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on $C(k,\ell)$ that improve on earlier bounds if $\ell \ge 2.75 k$.