Dynamic Graph Coloring

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 \(O(\mathcal {C} dN ^{1/d})\)-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 \(O(dN ^{1/d})\) 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 \(\varOmega (N ^\frac{2}{c(c-1)})\) vertices per update, for any constant \(c \ge 2\).