The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with $Δ+1$ colors, where $Δ$ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of $(Δ+1)$-coloring in the standard message passing model remains one of the most important open questions of the area. In this paper, we aim to improve our understanding of the deterministic complexity of $(Δ+1)$-coloring as a function of $Δ$ in a special family of graphs for which significantly faster algorithms are already known. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. In general, in graphs of neighborhood independence $θ=O(1)$ (e.g., line graphs), it is known that $(Δ+1)$-coloring can be solved in $2^{O(\sqrt{\logΔ})}+O(\log^* n)$ rounds. In the present paper, we significantly improve this result, and we show that in graphs of neighborhood independence $θ$, a $(Δ+1)$-coloring can be computed in $(θ\cdot\logΔ)^{O(\log\logΔ/ \log\log\logΔ)}+O(\log^* n)$ rounds and thus in quasipolylogarithmic time in $Δ$ as long as $θ$ is at most polylogarithmic in $Δ$. We also show that the known approach that leads to a polylogarithmic in $Δ$ algorithm for $(2Δ-1)$-edge coloring already fails for edge colorings of hypergraphs of rank at least $3$.

翻译：分布式染色问题无疑是分布式图算法领域研究的核心问题之一。该问题最标准的形式要求用$Δ+1$种颜色对图进行顶点正常染色，其中$Δ$为图的最大度数。尽管分布式环境下对染色问题已有大量研究工作，但在标准消息传递模型中确定$(Δ+1)$-染色问题的确定性复杂性，仍是该领域最重要的未解问题之一。本文旨在深化对一类特殊图中$(Δ+1)$-染色确定性复杂性的理解——这类图已有显著更快的算法，且其复杂性表示为$Δ$的函数。图的邻域独立性$θ$定义为图中任意节点的成对非相邻邻居的最大数量。一般而言，对于邻域独立性$θ=O(1)$的图（例如线图），已知$(Δ+1)$-染色可在$2^{O(\sqrt{\logΔ})}+O(\log^* n)$轮内求解。本文显著改进了这一结果，证明在邻域独立性为$θ$的图中，$(Δ+1)$-染色可在$(θ\cdot\logΔ)^{O(\log\logΔ/ \log\log\logΔ)}+O(\log^* n)$轮内完成，因此当$θ$至多为$Δ$的多对数函数时，算法在$Δ$上达到拟多对数时间。我们还证明，已知的可对$(2Δ-1)$-边染色实现$Δ$多对数时间的方法，在秩至少为3的超图的边染色中已失效。