arXiv 1401.65832014-01-25math.CO

格图的无线电数

The Radio Number of Grid Graphs

Tian-Shun Allan Jiang

备注：: 17 pages, 7 figures

摘要：

无线电数问题使用图论模型来模拟无线网络上的最佳频率分配。连通图$G$的无线电标号是函数$f:V(G) \to \mathbb Z_{0}^+$，使得对于每对顶点$u,v \in V(G)$，我们有$\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$，其中$\text{diam}(G)$表示$G$的直径，$d(u,v)$表示顶点$u$和$v$之间的距离。令$\text{span}(f)$为分配给$V(G)$的最大标签和最小标签之间的差。然后，图$\text{rn}(G)$的\textit{radio number}被定义为在$G$的所有无线电标记上的$\text{span}(f)$的最小值。到目前为止，关于网格图的无线电数的结果很少：2009年Calles和Gomez给出了正方形网格的上界和下界，2008年Flores和Lewis无法完全确定梯形图的无线电数（$n$网格为2）。在本文中，我们完全确定了无线电数的网格图$G_{a,b}$的$a,b>2$，特征的问题，并提供了一个封闭形式的解决方案。这些结果在无线网络，如蜂窝塔和环境传感器的无线电频率分配的优化的影响。

英文摘要：

The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have been few results on the radio number of the grid graph: In 2009 Calles and Gomez gave an upper and lower bound for square grids, and in 2008 Flores and Lewis were unable to completely determine the radio number of the ladder graph (a 2 by $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the problem and providing a closed-form solution to each. These results have implications in the optimization of radio frequency assignment in wireless networks such as cell towers and environmental sensors.

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。

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英文摘要：

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