Realizing degree sequences with $\mathcal S_3$-connected graphs

A graph $G$ is $\mathcal S_3$-connected if, for any mapping $\beta : V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} \beta(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv \beta(v)\pmod{3}$ for any $v \in V(G)$. It is known that $\mathcal S_3$-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 $\mathcal{S}_{3}$-connected realization: A graphic sequence $\pi=(d_1,\, \ldots,\, d_n )$ has an $\mathcal S_3$-connected realization if and only if $\min \{d_1,\, \ldots,\, d_n\} \ge 4$ and $\sum^n_{i=1}d_i \ge 6n - 4$. Consequently, every graphic sequence $\pi=(d_1,\, \ldots,\, d_n )$ with $\min \{d_1,\, \ldots,\, d_n\} \ge 6$ 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.