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arXiv 1609.03002math.CO
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树的无线电编号

Radio number of trees

Devsi Bantva, Samir Vaidya, Sanming Zhou

备注：
19 pages, 7 figures. This is the final version accepted in Discrete Applied Mathematics
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摘要：

图$G$的无线电标号是映射$f: V(G) \rightarrow \{0, 1, 2, \ldots\}$，使得对于$G$的每对不同顶点$u, v$，$|f(u)-f(v)|\geq d + 1 - d(u,v)$，其中$d$是$G$的直径，$d(u,v)$是$G$中$u$和$v$之间的距离。$G$的无线电编号是最小整数$k$，使得$G$具有带有$\max\{f(v) : v \in V(G)\} = k$的无线电标记$f$。我们给出了树的无线电数的下界要达到的一个充分必要条件，另外两个充分条件相同的约束要达到的树，和树的无线电数的上界。使用这些，我们确定了三个家庭的树的无线电号码。

英文摘要：

A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that $|f(u)-f(v)|\geq d + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$, where $d$ is the diameter of $G$ and $d(u,v)$ the distance between $u$ and $v$ in $G$. The radio number of $G$ is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.

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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