Fully Dynamic Algorithms for Coloring Triangle-Free Graphs

TL;DR

This paper introduces a randomized dynamic algorithm for maintaining an optimal coloring of triangle-free graphs with high probability and efficient amortized update time, utilizing a novel entropy compression analysis.

Contribution

It presents the first dynamic algorithm for $O(rac{ riangle}{ ext{ln} riangle})$ coloring of triangle-free graphs with amortized update time $ riangle^{o(1)} ext{log} n$, employing a new entropy compression technique.

Findings

Achieves $O(rac{ riangle}{ ext{ln} riangle})$ coloring in dynamic graphs.

Amortized update time is $ riangle^{o(1)} ext{log} n$ per operation.

Introduces a novel entropy compression analysis for dynamic algorithms.

Abstract

A celebrated result of Johansson in graph theory states that every triangle-free graph of maximum degree $\Delta$ can be properly colored with $O(\Delta/\ln\Delta)$ colors, improving upon the "greedy bound" of $\Delta+1$ coloring in general graphs. This coloring can also be found in polynomial time. We present an algorithm for maintaining an $O(\Delta/\ln\Delta)$ coloring of a dynamically changing triangle-free graph that undergoes edge insertions and deletions. The algorithm is randomized and on $n$-vertex graphs has amortized update time of $\Delta^{o(1)}\log{n}$ per update with high probability, even against an adaptive adversary. A key to the analysis of our algorithm is an application of the entropy compression method that to our knowledge is new in the context of dynamic algorithms. This technique appears general and is likely to find other applications in dynamic problems and thus can be of its own independent interest.