Abstract
The problem of on-line coloring of an arbitrary graphs is known to be hard. Here we consider the problem of on-line coloring in the simplified situation where the input graph is KKs,t-free. We show that the on-line coloring algorithm with the First Fit strategy of proposed by Capponi and Pilotto in [1] is the best one for KK1,t-free graphs (t≥3). A.Capponi and C.Pilotto have shown that this algorithm achieves a competitive ratio of t−1 while we show that it is the best possible. However for the family of KKs,t-free graphs (s≥2, t≥2) there exists no on-line coloring algorithm with a competitive function. The problem of an on-line cliques covering for these families is hard. There exists no on-line cliques covering algorithm with a competitive function for the family of KKs,t-free graphs (s≥ 1, t≥3). The additional assumption that the input graph is given in a connected way does not help a lot and does not change our results described above.
Similar content being viewed by others
References
Capponi, A., Pilloto, C.: On-line coloring and on-line partitioning of graphs. Manusricpt (2003)
Chrobak, M., Ślusarek, M.: On some packing problems relating to dynamical storage allocation. Informatique Theorique et Applications 22, 487–499 (1988)
Graham, R.L.: Bounds for certain multiprocessor anomalies. Bell System Technical Journal 45, 1563–1581 (1966)
Gyárfás, A., Lehel, J.: On-line and first fit colorings of graphs. Journal of Graph Theory 12, 217–227 (1988)
Gyárfás, A., Király, Z., Lehel, J.: Effective on-line coloring of P5-free} graphs. Combinatorica 11, 181–184 (1991)
Gyárfás, A., Lehel, J.: On-line 3-chromatic graphs I. Triangle-free graphs. SIAM Journal of Discrete Mathematics 12(3), 385–411 (1999)
Kierstead, H.A., Penrice, S.G., Trotter, W.T.: First-fit and on-line coloring of graphs which do not induce P5. SIAM Journal of Discrete Mathematics 8, 485–498 (1995)
Ślusarek, M.: Optimal on-line coloring of circular arc graphs. Informatique Thèorique et Applications 29, 423–429 (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cieślik, I. On-line coloring and cliques covering for KK s,t -free graphs. Acta Informatica 42, 1–20 (2005). https://doi.org/10.1007/s00236-005-0167-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00236-005-0167-1