Palette Sparsification via FKNP

TL;DR

This paper establishes a connection between spread distributions and palette sparsification, providing a new probabilistic framework for coloring graphs with bounded degree.

Contribution

It introduces a novel use of spread distributions to derive palette sparsification results for graphs with bounded maximum degree.

Findings

Existence of a $C/D$-spread distribution for $(D+1)$-colorings

Application of Frankston, Kahn, Narayanan, and Park's threshold-spread connection

Derivation of a palette sparsification theorem for bounded degree graphs

Abstract

A random set $S$ is $p$-spread if, for all sets $T$, $$\mathbb{P}(S \supseteq T) \leq p^{|T|}.$$ There is a constant $C>1$ large enough that for every graph $G$ with maximum degree $D$, there is a $C/D$-spread distribution on $(D+1)$-colorings of $G$. Making use of a connection between thresholds and spread distributions due to Frankston, Kahn, Narayanan, and Park, a palette sparsification theorem of Assadi, Chen, and Khanna follows.