European Journal of Combinatorics ( IF 1 ) Pub Date : 2020-08-27 , DOI: 10.1016/j.ejc.2020.103203 Angel Chavez , Daphne Der-Fen Liu , Mason Shurman
Let be a graph, and let be a positive integer. The radio--number of is the smallest integer for which there exists a function such that for any two vertices , , where is the distance between and . In particular, when is the diameter of , the radio--number is called the radio number of . This article contains four major parts. First, we extend Liu’s lower bound on radio numbers of trees to radio--numbers of trees. We call a tree whose radio--number is equal to this lower bound a -lower bound tree. Second, we establish properties of -lower bound trees, and apply these properties to obtain shorter proofs and generalizations of some known results. Third, we investigate the minimum integer such that a given tree is a -lower bound tree. Fourth, we define the joined union operation on trees, where trees are composed by merging a vertex from each tree into a single vertex. We use this operation to construct new -lower bound trees in three ways. Finally, we pose several questions for future study.
中文翻译:
最佳无线电-树木的标签
让 做个图,让 是一个正整数。该放射-个 是最小的整数 为此功能 这样对于任何两个顶点 , ,在哪里 是之间的距离 和 。特别是当 是的直径 , 收音机--number被称为无线电号码的。本文包含四个主要部分。首先,我们将刘的树的无线电数量下限扩展到无线电-树木数量。我们称一棵树,其无线电-number等于此下限a -下界树。其次,我们建立-下界树,并应用这些属性以获得较短的证明和某些已知结果的概括。第三,我们研究最小整数 这样一棵给定的树 是一个 -下界树。第四,我们在树上定义联合联合操作,其中树是通过将每棵树的顶点合并为单个顶点来组成的。我们使用此操作来构造新的下界树的三种方式。最后,我们提出一些问题,以供将来研究。