Palette Sparsification Beyond $(\Delta+1)$ Vertex Coloring
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 sublineartime 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 trianglefree 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 trianglefree 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 sublineartime algorithms.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.10456
 Bibcode:
 2020arXiv200610456A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics