The cycle discrepancy of three - regular graphs by Sarmad Abbasi 1 117 - BB , Phase IV , DHA Lahore 08903 Pakistan

Let G = (V, E) be an undirected graph and C(G) denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as cycdisc(G) = min χ:V !→{+1,−1} max C∈C(G) ∣∣∣∣ ∑ v∈C χ(v) ∣∣∣∣ . We show that, if G is a three-regular graph with n vertices, then cycdisc(G) ≤ 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) time a labeling χ, such that max C∈C(G) ∣∣∣∣ ∑ v∈C χ(v) ∣∣∣∣ ≤ n + 2 6 . Some interesting open problems regarding cycle discrepancy are also suggested.