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

Abstract

If F is a (possibly infinite) subset of an abelian group Γ, then we define $f(F,\Gamma )$ as the smallest natural number such that every $f(F,\Gamma )$-edge-connected (finite) graph G has a flow where all flow values are elements in F. We prove that $f(F,\Gamma )$ 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 $k⩾3$, every $(3k-1)$-edge-connected graph has a collection of k even subgraphs such that every edge is in precisely $k-1$ of them. Finally, the flow result gives a considerable freedom to prescribe the flow values in the $(2+ϵ)$-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, $1+1/k$. It also has, for any irrational number ϵ, a flow with flow values 1, $1+ϵ$, $1+ϵ+1/k$.