Asymptotics for Palette Sparsification

Abstract

It is shown that the following holds for each $\epsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$ and "lists" $L_v$ ($v\in V(G)$) chosen independently and uniformly from the ($(1+\epsilon )\ln n$)-subsets of $\{1,...,D+1\}$, \[ G \text{ admits a proper coloring } σ\text{ with } σ_v \in L_v \forall v \] with probability tending to 1 as $D\to \infty$. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.

29 pages