A randomized algorithm maintains O(Δ / ln Δ) coloring of dynamically changing triangle-free graphs with amortized Δ^{o(1)} log n update time per edge update against an adaptive adversary.
Combinatorics (Keszthely, 1976), Coll
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Fully Dynamic Algorithms for Coloring Triangle-Free Graphs
A randomized algorithm maintains O(Δ / ln Δ) coloring of dynamically changing triangle-free graphs with amortized Δ^{o(1)} log n update time per edge update against an adaptive adversary.