Group flow, complex flow, unit vector flow, and the (2+ϵ)(2+ϵ)-flow conjecture

If F is a (possibly infinite) subset of an abelian group Γ  , then we define f(F,Γ)f(F,Γ) as the smallest natural number such that every f(F,Γ)f(F,Γ)-edge-connected (finite) graph G has a flow where all flow values are elements in F  . We prove that f(F,Γ)f(F,Γ) exists if and only if some odd sum of elements in F   equals some even sum. We discuss various instances of this problem. We prove that every 6-edge-connected graph has a flow whose flow values are the three roots of unity in the complex plane. If the edge-connectivity 6 can be reduced, then it can be reduced to 4, and the 3-flow conjecture follows. We prove that every 14-edge-connected graph has a flow whose flow values are the five roots of unity in the complex plane. Any such flow is balanced modulo 5. So, if the edge-connectivity 14 can be reduced to 9, then the 5-flow conjecture follows, as observed by F. Jaeger. We use vector flow to prove that, for each odd natural number k⩾3k⩾3, every (3k−1)(3k−1)-edge-connected graph has a collection of k   even subgraphs such that every edge is in precisely k−1k−1 of them. Finally, the flow result gives a considerable freedom to prescribe the flow values in the (2+ϵ)(2+ϵ)-flow conjecture by L. Goddyn and P. Seymour. For example, if k is a natural number and G is a 6k-edge-connected graph, then G   has a flow with flow values 1, 1+1/k1+1/k. It also has, for any irrational number ϵ  , a flow with flow values 1, 1+ϵ1+ϵ, 1+ϵ+1/k1+ϵ+1/k.