Realizing degree sequences with S₃-connected graphs

A graph G is S₃-connected if, for any mapping β : V (G) ↦ Z₃ with ∑v∈ V(G) β(v)≡ 03, there exists a strongly connected orientation D satisfying d⁺D(v)-d⁻D(v)≡ β(v)3 for any v ∈ V(G). It is known that S₃-connected graphs are contractible configurations for the property of flow index strictly less than three. In this paper, we provide a complete characterization of graphic sequences that have an S₃-connected realization: A graphic sequence π=(d₁,\, …,\, dₙ ) has an S₃-connected realization if and only if min ₁,\, …,\, dₙ\ ≥ 4 and ∑ⁿᵢ₌₁dᵢ ≥ 6n - 4. Consequently, every graphic sequence π=(d₁,\, …,\, dₙ ) with min ₁,\, …,\, dₙ\ ≥ 6 has a realization G with flow index strictly less than three. This supports a conjecture of Li, Thomassen, Wu and Zhang [European J. Combin., 70 (2018) 164-177] that every $6$-edge-connected graph has flow index strictly less than three.