Handbook of combinatorics (vol. 2)
Approximation of Multi-color Discrepancy
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
Balanced Coloring: Equally Easy for All Numbers of Colors?
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Combinatorics, Probability and Computing
Hereditary Discrepancies in Different Numbers of Colors II
SIAM Journal on Discrete Mathematics
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In this article, we investigate the interrelation between the discrepancies of a given hypergraph in different numbers of colors. Being an extreme example we determine the multi-color discrepancies of the k-balanced hypergraph Hnk on partition classes of (equal) size n. Let c, k, n ∈ N. Set k0:=k mod c and bnkc := (n - [n/[c/k]])k/c. For the discrepancy in c colors we show bnk0c ≤ disc(Hnk, c) bnk0c + 1, if k0 ≠ 0, and disc(Hnk, c)= 0, if c divides k. This shows that, in general, there is little correlation between the discrepancies of Hnk in different numbers of colors. If c divides k though, disc(H,c) ≤ (k/c)disc(H,k) holds for any hypergraph H.