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 $c \ge 2$.

M. K. was partially supported by MEXT KAKENHI grant Nos. 12H00855, and 17K12635. A. v. R. and M. R. were supported by JST ERATO Grant Number JPMJER1305, Japan. L.B. was supported by the ETH Postdoctoral Fellowship. S. V. was partially supported by NSERC and the Carleton-Fields postdoctoral award.