Abstract
While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k ≥ 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d - 3, where d (the “dimension”) is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is at most max{2d-3, 2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley, New York, 1992. Chapter 12, pages 185–196.
Jin Akiyama and Jorge Urrutia. A note on balanced colourings for lattice points. Discrete Mathematics, 83(1):123–126, 1990.
Tetsuo Asano, Tomomi Matsui, and Takeshi Tokuyama. On the complexities of the optimal rounding problems of sequences and matrices. In Proceedings of the 7th Scandinavian Workshop on Algorithm Theory (SWAT’00), Bergen, Norway, July 2000. To appear.
József Beck. Some results and problems in “combinatorial discrepancy theory”. In Topics in Classical Number Theory: Proceedings of the International Conference on Number Theory, pages 203–218, Budapest, Hungary, July 1981. Appeared in Colloquia Mathematica Societatis János Bolyai, volume 34, 1994.
József Beck and Tibor Fiala. “Integer-making” theorems. Discrete Applied Mathematics, 3:1–8, 1981.
József Beck and Vera T. Sós. Discrepancy theory. In Handbook of Combinatorics, volume 2, pages 1405–1446. Elsevier, Amsterdam, 1995.
A.J.W. Hilton and D. de Werra. A sufficient condition for equitable edge-colourings of simple graphs. Discrete Mathematics, 128(1–3): 179–201, 1994.
Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 216–226, San Diego, California, May 1978.
Jiří Šíma. Table rounding problem. Comput. Artificial Intelligence, 18(3): 175–189, 1999.
Joel Spencer. Geometric discrepancy theory. Contemporary Mathematics, 223, 1999.
Joel Spencer. Personal communication. 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Biedl, T.C. et al. (2000). Balanced k-Colorings. In: Nielsen, M., Rovan, B. (eds) Mathematical Foundations of Computer Science 2000. MFCS 2000. Lecture Notes in Computer Science, vol 1893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44612-5_16
Download citation
DOI: https://doi.org/10.1007/3-540-44612-5_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67901-1
Online ISBN: 978-3-540-44612-5
eBook Packages: Springer Book Archive