Palette Sparsification Beyond $(\Delta+1)$ Vertex Coloring

TL;DR

This paper advances palette sparsification techniques for graph coloring, reducing the number of colors needed for proper coloring in various graph classes and models, with implications for efficient algorithms.

Contribution

It introduces new bounds on the number of colors required for palette sparsification in different graph coloring scenarios, extending previous results.

Findings

Sampling $O_{ ext{ε}}}(rac{1}{ ext{ε}} \, ext{log} n)$ colors suffices for $(1+ε) \, ext{Δ}$ coloring.

Sampling $O(\Delta^{γ} + \, ext{√log} n)$ colors suffices for proper coloring of triangle-free graphs.

Sampling $O_{ ext{ε}}}( ext{log} n)$ colors suffices for proper coloring with list constraints.

Abstract

A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $\Delta$, sampling $O(\log{n})$ colors per each vertex independently from $\Delta+1$ colors almost certainly allows for proper coloring of $G$ from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for $(\Delta+1)$ coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. In this paper, we further study palette sparsification problems: * We prove that for $(1+\varepsilon) \Delta$ coloring, sampling only $O_{\varepsilon}(\sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper coloring from the sampled colors. * A natural family of graphs with chromatic number much smaller than $(\Delta+1)$ are triangle-free graphs which are $O(\frac{\Delta}{\ln{\Delta}})$ colorable. We prove that sampling $O(\Delta^{\gamma} + \sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper $O_{\gamma}(\frac{\Delta}{\ln{\Delta}})$ coloring of triangle-free graphs. * We show that sampling $O_{\varepsilon}(\log{n})$ colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of $(1+\varepsilon) \cdot deg(v)$ arbitrary colors, or even only $deg(v)+1$ colors when the lists are the sets $\{1,\ldots,deg(v)+1\}$. Similar to previous work, our new palette sparsification results naturally lead to a host of new and/or improved algorithms for vertex coloring in different models including streaming and sublinear-time algorithms.