Asymptotics for Palette Sparsification from Variable Lists

TL;DR

The paper proves that for large maximum degree graphs, randomly selecting small subsets from lists allows proper coloring with high probability, extending and optimizing previous palette sparsification results.

Contribution

It establishes an asymptotically optimal palette sparsification theorem for graphs with large maximum degree, improving prior bounds and methods.

Findings

Proper coloring is achievable with high probability using small random subsets.

The result generalizes and optimizes previous palette sparsification theorems.

Asymptotic optimality is demonstrated for large maximum degree graphs.

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.