Abstract
For an abelian group \Gamma , a graph G is said to be \Gamma-flow-critical if G does not admit a nowhere-zero \Gamma-flow, but for each edge e \in E(G), the contraction G/e has a nowhere-zero \Gamma-flow. We obtain a bound on the density of Z3-flow-critical graphs drawn on a fixed surface, generalizing the planar case of the bound on the density of 4-critical graphs by Kostochka and Yancey.