On mod (2p+1)(2p+1)-orientations of graphs

It is shown that every (2p+1)log2(|V(G)|)(2p+1)log2(|V(G)|)-edge-connected graph G   has a mod (2p+1)(2p+1)-orientation, and that a (4p+1)(4p+1)-regular graph G   has a mod (2p+1)(2p+1)-orientation if and only if V(G)V(G) has a partition (V+,V−)(V+,V−) such that ∀U⊆V(G)∀U⊆V(G),|∂G(U)|⩾(2p+1)||U∩V+|−|U∩V−||.|∂G(U)|⩾(2p+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.