Nowhere-zero 3-flows in products of graphs

A graph G is an odd-circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere-zero 3-flow unless G is an odd-circuit tree and H has a bridge. This theorem is a partial result to the Tutte’s 3-flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere-zero 3-flow. A byproduct of this theorem is that every bridgeless Cayley graph G 1⁄4 Cayð ;SÞ on an abelian group with a minimal generating set S admits a nowhere-zero 3-flow except for odd prisms. 2005 Wiley Periodicals, Inc. J Graph Theory 50: 79–89, 2005