Dynamic Graph Coloring

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(𝒞 dN ^1/d) -coloring while recoloring at most O ( d ) vertices per update, where 𝒞 and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(𝒞 d) -coloring with O(dN ^1/d) recolorings per update. The two converge when d = log N , maintaining an O(𝒞log 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 (N ^2/c(c-1)) vertices per update, for any constant c ≥ 2 .