Graphs and Combinatorics

, Volume 27, Issue 1, pp 27–46

The Cycle Discrepancy of Three-Regular Graphs

Original Paper

Abstract

Let G = (V, E) be an undirected graph and $${{\mathcal C}(G)}$$ denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as
$${\rm cycdisc}(G) = \min_{\chi: V \mapsto \{+1, -1\}} \max_{ C \in {\mathcal C} (G)} \left|\sum_{v \in C} \chi(v)\right|.$$
We show that, if G is a three-regular graph with n vertices, then
$${\rm cycdisc}(G) \leq \frac{n +2}{6}.$$
This bound is best possible and is achieved by very simple graphs. Our proof is algorithmic and allows us to compute in O(n 2) time a labeling χ, such that
$$\max_{ C \in \mathcal{C}(G)} \left| \sum_{v \in C} \chi(v) \right| \leq \frac{n + 2}{6}.$$
Some interesting open problems regarding cycle discrepancy are also suggested.

Keywords

Graph coloring Discrepancy Cycle discrepancy

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References

1. 1.
Beck J., Chen W.L.: Irregularities of Distribution. Cambridge University Press, New York (1987)
2. 2.
Beck J., Fiala W.: Integer making theorems. Discrete Appl. Math. 3, 1–8 (1981)
3. 3.
Bollobás B.: Modern Graph Theory. Springer-Verlag, Berlin (1998)
4. 4.
Brooks R.L.: On coloring the nodes of a network. Proc. Cambr. Philos. Soc. 37, 194–197 (1941)
5. 5.
Chazelle B.: The Discrepancy Method: Randomness and Complexity. Cambridge University Press, New York (2000)
6. 6.
Kranakis E., Krizanc D., Ruf B., Urrutia J., Woeginger G.: The vc-dimension of set systems defined by graphs. Discrete Appl. Math. 77(3), 237–257 (1997)
7. 7.
Matousek J.: Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics. Springer, Berlin (1999)Google Scholar
8. 8.
Schaefer M.: Deciding the Vapnik-Chervonenkis dimension is $${\Sigma_p^3}$$-complete. J. Comput. Syst. Sci. 58(1), 177–182 (1999)
9. 9.
Spencer J.: Six standard deviations suffice. Trans. Am. Math. Soc. 289(2), 679–706 (1985) 