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arXiv · 2306.00171

Asymptotics for Palette Sparsification

Jeff Kahn·Charles Kenney
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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 $\{1, ..., D+1\}$, \[ G \text{ admits a proper coloring } σ\text{ with } σ_v \in L_v \forall v \] 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.

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Jeff Kahn, Charles Kenney. 2023-05-31. Asymptotics for Palette Sparsification. https://arxiv.org/abs/2306.00171

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