arXiv 1708.050412017-08-16math.CO

无爪三次图的全强迫与零强迫

Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

Randy Davila, Michael Henning

摘要：

图$G$的顶点的动态着色从着色顶点的初始子集$S$开始，所有剩余的顶点是未着色的。在每个离散时间间隔，一个着色顶点与恰好一个非着色邻居迫使这个非着色邻居着色。初始集合$S$被称为$G$的强制集合（迫零集合），如果通过迭代地应用强制过程，$G$中的每个顶点都被着色。如果初始集$S$具有导出$G$的无孤立点子图的附加性质，则称$S$为$G$的全强制集. $G$的总强迫数，记作$F_t(G)$，是$G$中总强迫集的最小基数。本文证明了：如果$G$是一个阶为~ $n$的连通三次图，且其生成$2$ -因子由三角形组成，则$F_t(G) \le \frac{1}{2}n$ .更一般地，我们证明了，如果$G$是一个连通的，无爪，立方图的顺序~ $n \ge 6$，然后$F_t(G) \le \frac{1}{2}n$，其中无爪图是一个图，不包含$K_{1,3}$作为诱导子图。在这些界限实现平等的图的特点。

英文摘要：

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set (zero forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The total forcing number of $G$, denoted $F_t(G)$, is the minimum cardinality of a total forcing set in $G$. We prove that if $G$ is a connected cubic graph of order~$n$ that has a spanning $2$-factor consisting of triangles, then $F_t(G) \le \frac{1}{2}n$. More generally, we prove that if $G$ is a connected, claw-free, cubic graph of order~$n \ge 6$, then $F_t(G) \le \frac{1}{2}n$, where a claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. The graphs achieving equality in these bounds are characterized.

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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