Dynamic Graph Coloring
Luis Barba, Jean Cardinal, Matias Korman, Stefan Langerman, André van Renssen, Marcel Roeloffzen, Sander Verdonschot · Lecture notes in computer science · 2017
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 $$c \ge 2$$ .