arXiv 1407.35312014-07-14math.CO

度序列作为$Z_3$ -连通图的实现

Realizing degree sequences as $Z_3$-connected graphs

Fan Yang, Xiangwen Li, Hong -Jian Lai

摘要：

一个整数值序列$π=(d_1, \ldots, d_n)$是{ \em graphic}，如果存在一个简单图$G$，其度序列为$π$ .我们说$π$有一个实现$G$。设$Z_3$是一个三阶循环群。一个图$G$是{ \em $Z_3$ -连通的}，如果对于每个映射$b:V(G)\to Z_3$使得$\sum_{v\in V(G)}b(v)=0$，有一个$G$的方向和一个映射$f: E(G)\to Z_3-\{0\}$使得对于每个顶点$v\in V(G)$，所有离开$v$的边上的$f$的值之和减去所有到达$v$的边上的$f$的值之和等于$b(v)$。如果一个整数值序列$π$有一个$Z_3$ -连通的实现$G$，则$π$有一个{ \em $Z_3$ -连通的实现} $G$。设$π=(d_1, \ldots, d_n)$是具有$d_1\ge \ldots \ge d_n\ge 3$的图形序列。本文证明了：如果$d_1\ge n-3$，则任一$π$都有$Z_3$ -连通实现，除非序列是$(n-3, 3^{n-1})$或$(k, 3^k)$或$(k^2, 3^{k-1})$，其中$k=n-1$和$n$是偶数;如果$d_{n-5}\ge 4$，则任一$π$都有$Z_3$ -连通实现，除非序列是$(5^2, 3^4)$或$(5, 3^5)$ .

英文摘要：

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 $(5, 3^5)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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