Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth

We deterministically compute a ∆+1 coloring in time O(∆5c+2 ·(∆5)/(∆1) + (∆1) + log∗ n) and O(∆5c+2 · (∆5)/∆ + ∆ + (∆5) log∆5 log n) for arbitrary constants d, and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i for a node and ∆ := ∆1. Our greedy algorithm improves the state-of-the-art ∆+1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If ∆ ∈ Ω(log log ∗ n n) and χ ∈ O(∆/ log log ∗ n n) then our algorithm executes in time O(logχ + log∗ n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest ∆ + 1 coloring algorithm running in time O(log∆ + √ log n). The algorithm works without knowledge of χ and uses less than ∆ colors, i.e., (1 − 1/O(χ))∆ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account.