# Coloring inductive graphs on-line

• Published:

## 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 maxG ∈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.

This is a preview of subscription content, log in via an institution to check access.

## Subscribe and save

Springer+ Basic
\$34.99 /Month
• Get 10 units per month
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

## References

1. Chaitin, C. G., Register allocation and spilling via graph coloring,Proc. Sigplan Symposium on Computer Construction, June 1982, Note 17.6, pp. 98–105.

2. Chung, F. R. K., Graham, R., Saks, M. E., A dynamic location problem for graphs,Combinatorica,9(2) (1989), 111–131.

3. Faigle, U., Kern, W., Turan, G., On the performance of on-line algorithms for partitioning problems,Acta Cybernet.,9 (1989), 107–119.

4. Gyarfas, A., Lehel, J., On-line and First Fit colorings of graphsJ. Graph Theory,12(2) (1988), 217–227.

5. Lovasz, L., Naor, M, Newman, I., Wigderson, A., Search problems in the decision tree model,Proc. 32nd Symposium on the Foundations of Computing Science, 1991.

6. Lovasz, L., Saks, M. E., Trotter, W. A., An on-line graph coloring algorithm with sublinear performance ratio,Discrete Math., Special Volume on Graph Theory and Combinatorics (1988), 319–326.

7. Karloff, H., Personal communication.

8. Kierstead, H. A., The linearity of First-Fit coloring of interval graphs,SIAM J. Discrete Math.,1(4) (1988), 526–530.

9. Kierstead, H. A., Trotter, W. A., An extremal problem in recursive combinatorics,Congress. numer.,33 (1981), 143–153.

10. Patterson, D. A., Reduced instruction set computers.Comm. CACM,28(1) (1985).

11. Saks, M., Personal communication.

12. Sleator, D. D., Tarjan, R. E., Amortized efficiency of list update and paging rules,Comm. CACM,28(2) (1985).

13. Personal communication, transmitted through L. Lovasz, M. E. Saks, and W. A. Trotter.

14. Vishwanathan, S., Randomized online coloring of graphs,Proc. 31st Annual Symposium on the Foundations of Computer Science, October 1990.

## Author information

### Authors and Affiliations

Authors

Communicated by Probhakar Raghavan.

This research was supported by an IBM Graduate Fellowship.

## Rights and permissions

Reprints and permissions

Irani, S. Coloring inductive graphs on-line. Algorithmica 11, 53–72 (1994). https://doi.org/10.1007/BF01294263