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Graphs and Combinatorics

, Volume 27, Issue 1, pp 27–46 | Cite as

The Cycle Discrepancy of Three-Regular Graphs

  • Sarmad AbbasiEmail author
  • Laeeq Aslam
Original Paper
  • 84 Downloads

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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Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.LahorePakistan
  2. 2.Punjab University College of Information Technology, University of the PunjabLahorePakistan

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