Integer flows and cycle covers

Results related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection %Y of cycles of G which covers all edges of G; U is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of V. By using Seymour’s nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by 10 even subgraphs (an even graph is one in which each vertex is of even degree). This result together with the cycle 4-cover theorem implies that every bridgeless graph has a cycle m-cover for any even number m z 4. We also prove that every graph with a nowhere-zero 4-flow has a cycle cover V such that the sum of lengths of the cycles in V is at most [E(G)1 + IV(G)1 -2, unless G belongs to a very special class of graphs.