# Symmetry breaking depending on the chromatic number or the neighborhood growth

## Abstract

We deterministically compute a Δ+1 coloring and a maximal independent set(MIS) in time O(Δ1/2+Θ(1/√h)+log*n) for Δ1+i≤Δ1+i/h, where Δj is defined as the maximal number of nodes within distance j for a node and Δ:=Δ1. Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(Δ+log*n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h≥36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O(logχ+log*n) for Δ∈Ω(log1+1/log*nn) and χ∈O(Δ/log1+1/log*nn). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest Δ+1 coloring algorithm running in time O(logΔ+√log n). The algorithm works without knowledge of χ and uses less than Δ colors, i.e., (1-1/O(χ))Δ 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. We also improve on the state of the art deterministic computation of (2,c)-ruling sets. © 2012 Elsevier B.V. All rights reserved.