# Coloring inductive graphs on-line

## Authors

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DOI: 10.1007/BF01294263

- Cite this article as:
- Irani, S. Algorithmica (1994) 11: 53. doi:10.1007/BF01294263

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## 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. Let*A(G)* be the number of colors used by algorithm*A* on a graph*G* and let*x(G)* be the chromatic number of*G*. 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 of*d*-inductive graphs. A graph*G* is*d*-inductive if the nodes of*G* can be numbered so that each node has at most*d* edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs are*x(G*)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that if*G* is*d*-inductive, then First Fit uses*O(d* log*n)* colors on*G*. This yields an upper bound of*o*(log*n*) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm for*d*-inductive graphs: we show that, for any*d* and any on-line graph coloring algorithm*A*, there is a*d*-inductive graph that forces*A* to use Ω(*d* log*n*) colors to color*G*. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookahead*l*, if it must color node*i* after examining only the first*l+i* nodes. We show that, for*l<n*/log*n*, the lower bound of*d* log*n* colors still holds.