On mod -orientations of graphs

It is shown that every ( 2 p + 1 ) log 2 ( | V ( G ) | ) -edge-connected graph G has a mod ( 2 p + 1 ) -orientation, and that a ( 4 p + 1 ) -regular graph G has a mod ( 2 p + 1 ) -orientation if and only if V ( G ) has a partition ( V + , V − ) such that ∀ U ⊆ V ( G ) , | ∂ G ( U ) | ⩾ ( 2 p + 1 ) | | U ∩ V + | − | U ∩ V − | | . These extend former results by Da Silva and Dahab on nowhere zero 3-flows of 5-regular graphs, and by Lai and Zhang on highly connected graphs with nowhere zero 3-flows.