Fully Dynamic Algorithms for Coloring Triangle-Free Graphs

2026-04-22

AI Summary

This work addresses the problem of efficiently maintaining sparse colorings in triangle-free graphs subject to edge insertions and deletions. The paper proposes a randomized dynamic algorithm that, for the first time, incorporates the entropy compression method into the analysis of dynamic graph algorithms. Under an adaptive adversary, the algorithm achieves— with high probability—an $O(\Delta / \log \Delta)$-coloring, where $\Delta$ denotes the maximum degree, with an amortized update time of $\Delta^{o(1)} \log n$. This result substantially improves upon the classical greedy bound of $\Delta + 1$ colors and introduces novel techniques that offer stronger theoretical guarantees for dynamic graph coloring.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A celebrated result of Johansson in graph theory states that every triangle-free graph of maximum degree $Δ$ can be properly colored with $O(Δ/\lnΔ)$ colors, improving upon the "greedy bound" of $Δ+1$ coloring in general graphs. This coloring can also be found in polynomial time. We present an algorithm for maintaining an $O(Δ/\lnΔ)$ 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 $Δ^{o(1)}\log{n}$ 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.

Problem

• dynamic graph coloring
• triangle-free graphs
• edge insertions and deletions
• proper coloring
• maximum degree

Innovation

• dynamic graph algorithms
• triangle-free coloring
• entropy compression
• randomized algorithms
• amortized update time