Palette Sparsification Beyond (Δ+1) Vertex Coloring
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A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every n-vertex graph G with maximum degree Δ, sampling O(logn) colors per each vertex independently from Δ+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 (Δ+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+ε) Δ coloring, sampling only O_ε(√(logn)) 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 (Δ+1) are triangle-free graphs which are O(Δ/lnΔ) colorable. We prove that sampling O(Δ^γ + √(logn)) colors per vertex is sufficient and necessary to obtain a proper O_γ(Δ/lnΔ) coloring of triangle-free graphs. * We show that sampling O_ε(logn) colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of (1+ε) · deg(v) arbitrary colors, or even only deg(v)+1 colors when the lists are the sets {1,…,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.
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