Nowhere‐zero 3‐flow and Z3‐connectedness in graphs with four edge‐disjoint spanning trees

Given a zero‐sum function β:V(G)→Z3 with ∑v∈V(G)β(v)=0, an orientation D of G with dD+(v)−dD−(v)=β(v) in Z3 for every vertex v∈V(G) is called a β‐orientation. A graph G is Z3‐connected if G admits a β‐orientation for every zero‐sum function β. Jaeger et al. conjectured that every 5‐edge‐connected graph is Z3‐connected. A graph is ⟨Z3⟩‐extendable at vertex v if any preorientation at v can be extended to a β‐orientation of G for any zero‐sum function β. We observe that if every 5‐edge‐connected essentially 6‐edge‐connected graph is ⟨Z3⟩‐extendable at any degree five vertex, then the above‐mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lovász et al., we prove that every graph with at least four edge‐disjoint spanning trees is Z3‐connected. Consequently, every 5‐edge‐connected essentially 23‐edge‐connected graph is ⟨Z3⟩‐extendable at any degree five vertex.