Asymptotics for Palette Sparsification from Variable Lists

Abstract: It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ of size $D+1$ (for $v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$ (independent of other choices), [ \mbox{$G$ admits a proper coloring $\sigma$ with $\sigma_v\in L_v$ $\forall v$} ] with probability tending to 1 as $D\to \infty$. When each $S_v $ is ${1\dots D+1}$, this is an asymptotically optimal version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna that was proved in an earlier paper by the present authors.