All 4-Edge-Connected HHD-Free Graphs are {mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called mathbbZ₃{mathbb{Z}_3}-connected if for all b: V ? mathbbZ₃{b: V rightarrow mathbb{Z}_3} with ?_{v ? V}b(v)=0{sum_{v in V}b(v)=0}, an orientation D = (V, A) of G has a mathbbZ₃{mathbb{Z}_3}-valued nowhere-zero flow f: A? mathbbZ₃-{0}{f: Arightarrow mathbb{Z}_3-{0}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){sum_{e in delta^+(v)}f(e)-sum_{e in delta^-(v)}f(e)=b(v)} for all v ? V{v in V}. We show that all 4-edge-connected HHD-free graphs are mathbbZ₃{mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the mathbbZ₃{mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.