Trade-offs in dynamic coloring for bipartite and general graphs

TL;DR

This paper explores the trade-offs in dynamic graph coloring algorithms, presenting new deterministic algorithms with improved update times and recoloring bounds for bipartite and general graphs in incremental and fully dynamic settings.

Contribution

It introduces novel deterministic algorithms for dynamic graph coloring with optimized update times, recoloring bounds, and applicability to bipartite and general graphs, advancing the state of the art.

Findings

Deterministic fully dynamic 2-coloring algorithm with O(log^2 n) amortized update time.

Incremental 2-coloring algorithm with O(log n) amortized update time and recolorings.

Algorithms for general graphs with O(√m) and O(γ + log n) update times for (Δ+1)-coloring.

Abstract

We present trade-offs in the incremental and fully dynamic settings to maintian a proper coloring. For any fully dynamic $2$-coloring algorithm, the maximum of the update time, number of recolorings, and query time is $\Omega(\log n)$. We present a deterministic fully dynamic $2$-coloring algorithm with $O(\log^2 n)$ amortized update time, $O(\log n)$ amortized query time, and one recoloring in the worst case. For any incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, the amortized update time and the amortized number of recolorings is $\Omega(\log n)$. For such an algorithm, in the worst case the update time and the number of recolorings is $\Omega(n)$. We then design a deterministic incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, with amortized $O(\log n)$ update time and amortized $O(\log n)$ many recolorings. Further, in the worst case the update time and the number of recolorings is $O(n)$. Further, we present a deterministic incremental $(1+2 \log n)$-coloring algorithm which explicitly maintains the color of every vertex after each update, with $O(\alpha(n))$ amortized update time, at most one recoloring and $O(1)$ query time. We then show a deterministic incremental $2$-coloring algorithm which does not maintain color of every vertex after each update, with amortized $O(\alpha(n))$ update time, amortized $O(\alpha(n))$ recolorings, and amortized $O(\alpha(n))$ query time. For general graphs and graphs of bounded arboricity $\gamma$ and maximum degree $\Delta$ we present a deterministic $(\Delta+1)$-coloring algorithm with $O(\sqrt{m})$ update time, $O(1)$ query time, and one recoloring. Finally, we show a deterministic $(\Delta+1)$-coloring algorithm with amortized $O(\gamma + \log{n})$ update time, $O(1)$ query time, and one recoloring.