Asymptotics for Palette Sparsification

TL;DR

The paper proves that for large maximum degree graphs, randomly chosen small color lists almost surely suffice for proper coloring, extending and optimizing previous palette sparsification results.

Contribution

It provides an asymptotically optimal probabilistic result for palette sparsification in graph coloring, improving prior bounds.

Findings

Proper coloring with small random lists is almost surely possible for large D.

The result is asymptotically optimal compared to previous theorems.

The probability tends to 1 as D approaches infinity.

Abstract

It is shown that the following holds for each $\varepsilon>0$. For $G$ an $n$-vertex graph of maximum degree $D$ and "lists" $L_v$ ($v \in V(G)$) chosen independently and uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $\{1, ..., D+1\}$, \[ G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v \] 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.