Coloring inductive graphs on-line

Abstract

In this paper we consider the problem of on-line graph coloring. In an instance of on-line graph coloring, the nodes are presented one at a time. As each node is presented, its edges to previously presented nodes are also given. Each node must be assigned a color, different from the colors of its neighbors, before the next node is given. LetA(G) be the number of colors used by algorithmA on a graphG and letx(G) be the chromatic number ofG. The performance ratio of an on-line graph coloring algorithm for a class of graphsC is max_{G ∈C(A(G)/χ(G))}. We consider the class ofd-inductive graphs. A graphG isd-inductive if the nodes ofG can be numbered so that each node has at mostd edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs arex(G)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that ifG isd-inductive, then First Fit usesO(d logn) colors onG. This yields an upper bound ofo(logn) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm ford-inductive graphs: we show that, for anyd and any on-line graph coloring algorithmA, there is ad-inductive graph that forcesA to use Ω(d logn) colors to colorG. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookaheadl, if it must color nodei after examining only the firstl+i nodes. We show that, forl<n/logn, the lower bound ofd logn colors still holds.