Tutte′s 3-Flow Conjecture and Short Cycle Covers

In this paper we prove: (i) If a graph G has a nowhere-zero 6-flow φ such that | Eodd (φ)| ≥ 23 |E(G)|, then G has a cycle cover in which the sum of the lengths of the cycles in the cycle cover is at most 4427 |E(G)|, where Eodd (φ) = {e ∈ E(G) : is odd}; (ii) if Tutte′s 3-Flow Conjecture is true, then every bridgeless graph G has a nowhere-zero 6-flow φ such that | Eodd (φ)| ≥ 23 | E(G)|.