All 4-Edge-Connected HHD-Free Graphs are Z3 -Connected

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• An undirected graph G = (V,E) is called Z3-connected if for all b : V → Z3 with Σv∈V b(v) = 0, an orientation D = (V,A) of G has a Z3-valued nowhere-zero flow f : A → Z3 − {0} such that Σe∈δ+(v) f(e)−Σe∈δ−(v)f(e) = b(v) for all v ∈ V . We show that all 4-edge-connected HHD-free graphs are Z3-connected. This extends the result due to Lai (2000), which proves the Z3-connectivity for 4-edge-connected chordal graphs.

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