Dynamic Graph Coloring

TL;DR

This paper introduces two dynamic graph coloring algorithms that balance the number of recolorings and colors used, along with a lower bound showing the limits of such algorithms for maintaining proper colorings.

Contribution

The paper presents novel algorithms for dynamic graph coloring with adjustable trade-offs and establishes a lower bound on recoloring complexity for maintaining proper colorings.

Findings

Algorithms achieve different trade-offs between recolorings and colors.

Converging algorithm maintains $O( ext{chromatic number} imes ext{log } N)$-coloring with $O( ext{log } N)$ recolorings.

Lower bound shows minimum recolorings needed for maintaining proper colorings in 2-colorable graphs.

Abstract

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any $d>0$, the first algorithm maintains a proper $O(\mathcal{C} d N^{1/d})$-coloring while recoloring at most $O(d)$ vertices per update, where $\mathcal{C}$ and $N$ are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an $O(\mathcal{C} d)$-coloring with $O(d N^{1/d})$ recolorings per update. The two converge when $d = \log N$, maintaining an $O(\mathcal{C} \log N)$-coloring with $O(\log N)$ recolorings per update. We also present a lower bound, showing that any algorithm that maintains a $c$-coloring of a $2$-colorable graph on $N$ vertices must recolor at least $\Omega(N^\frac{2}{c(c-1)})$ vertices per update, for any constant $c \geq 2$.