Realizing degree sequences as $Z_3$-connected graphs

An integer-valued sequence $π=(d_1, \ldots, d_n)$ is {\em graphic} if there is a simple graph $G$ with degree sequence of $π$. We say the $π$ has a realization $G$. Let $Z_3$ be a cyclic group of order three. A graph $G$ is {\em $Z_3$-connected} if for every mapping $b:V(G)\to Z_3$ such that $\sum_{v\in V(G)}b(v)=0$, there is an orientation of $G$ and a mapping $f: E(G)\to Z_3-\{0\}$ such that for each vertex $v\in V(G)$, the sum of the values of $f$ on all the edges leaving from $v$ minus the sum of the values of $f$ on the all edges coming to $v$ is equal to $b(v)$. If an integer-valued sequence $π$ has a realization $G$ which is $Z_3$-connected, then $π$ has a {\em $Z_3$-connected realization} $G$. Let $π=(d_1, \ldots, d_n)$ be a graphic sequence with $d_1\ge \ldots \ge d_n\ge 3$. We prove in this paper that if $d_1\ge n-3$, then either $π$ has a $Z_3$-connected realization unless the sequence is $(n-3, 3^{n-1})$ or is $(k, 3^k)$ or $(k^2, 3^{k-1})$ where $k=n-1$ and $n$ is even; if $d_{n-5}\ge 4$, then either $π$ has a $Z_3$-connected realization unless the sequence is $(5^2, 3^4)$ or $(5, 3^5)$.

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