arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1608.00747math.CO
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图的迫零数的几个界

Some Bounds on the Zero Forcing Number of a Graph

Michael Gentner, Dieter Rautenbach

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摘要：

一个图$G$的顶点集$Z$是$G$的迫零集，如果最初用$1$标记$Z$中的所有顶点，并用$0$标记$G$中的所有剩余顶点，然后，迭代地并尽可能长地，如果$u$是具有标签$1$的某个顶点的唯一具有标签$0$的邻居，则将某个顶点$u$的标签从$0$改变为$1$，产生$G$的整个顶点集。迫零数$Z(G)$定义为$G$的迫零集的最小阶数，被提出作为与$G$相关的矩阵的余秩的上界，也被认为与量子物理和逻辑电路有关。考虑到迫零数的计算难度，上界和下界是有意义的。对Amos，Caro，Davila和Pepper的结果进行了改进，证明了对于阶为$n$且最大度为$Δ$的连通图$G$，$Z(G)\leq \frac{Δ-2}{Δ-1}n$至少为$3$当且仅当$G$不属于$\{ K_{Δ+1},K_{Δ,Δ},K_{Δ-1,Δ},G_1,G_2\}$，其中$G_1$和$G_2$分别是阶为$5$和$7$的两个特殊图.对于阶为$n$，最大度为$3$，围长至少为$5$的连通图$G$，我们证明了$Z(G)\leq \frac{n}{2}-Ω\left(\frac{n}{\log n}\right)$ .利用概率论的方法，证明了对于$r$ -正则图$G$的$Z(G)\leq \left(1-\frac{H_r}{r}+o\left(\frac{H_r}{r}\right)\right)n$，$G$的阶为$n$，围长至少为$5$，其中$H_r$是$r$ -阶调和数.最后，我们证明了围长为$g\in \{ 5,6\}$且度为最小$δ$的图$G$的$Z(G)\geq (g-2)(δ-2)+2$，部分地证实了Davila和Kenter的一个猜想.

英文摘要：

A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if initially labeling all vertices in $Z$ with $1$ and all remaining vertices of $G$ with $0$, and then, iteratively and as long as possible, changing the label of some vertex $u$ from $0$ to $1$ if $u$ is the only neighbor with label $0$ of some vertex with label $1$, results in the entire vertex set of $G$. The zero forcing number $Z(G)$, defined as the minimum order of a zero forcing set of $G$, was proposed as an upper bound of the corank of matrices associated with $G$, and was also considered in connection with quantum physics and logic circuits. In view of the computational hardness of the zero forcing number, upper and lower bounds are of interest. Refining results of Amos, Caro, Davila, and Pepper, we show that $Z(G)\leq \frac{Δ-2}{Δ-1}n$ for a connected graph $G$ of order $n$ and maximum degree $Δ$ at least $3$ if and only if $G$ does not belong to $\{ K_{Δ+1},K_{Δ,Δ},K_{Δ-1,Δ},G_1,G_2\}$, where $G_1$ and $G_2$ are two specific graphs of orders $5$ and $7$, respectively. For a connected graph $G$ of order $n$, maximum degree $3$, and girth at least $5$, we show $Z(G)\leq \frac{n}{2}-Ω\left(\frac{n}{\log n}\right)$. Using a probabilistic argument, we show $Z(G)\leq \left(1-\frac{H_r}{r}+o\left(\frac{H_r}{r}\right)\right)n$ for an $r$-regular graph $G$ of order $n$ and girth at least $5$, where $H_r$ is the $r$-th harmonic number. Finally, we show $Z(G)\geq (g-2)(δ-2)+2$ for a graph $G$ of girth $g\in \{ 5,6\}$ and minimum degree $δ$, which partially confirms a conjecture of Davila and Kenter.

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