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Optimal radio-k-labelings of trees
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 G be a graph, and let k be a positive integer. The radio-k-number of G is the smallest integer s for which there exists a function f:V(G){0,1,2,,s} such that for any two vertices u,vV(G), |f(u)f(v)|k+1d(u,v), where d(u,v) is the distance between u and v. In particular, when d is the diameter of G, the radio-d-number is called the radio number of G. This article contains four major parts. First, we extend Liu’s lower bound on radio numbers of trees to radio-k-numbers of trees. We call a tree whose radio-k-number is equal to this lower bound a k-lower bound tree. Second, we establish properties of k-lower bound trees, and apply these properties to obtain shorter proofs and generalizations of some known results. Third, we investigate the minimum integer k such that a given tree T is a k-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 k-lower bound trees in three ways. Finally, we pose several questions for future study.



中文翻译:

最佳无线电ķ-树木的标签

G 做个图,让 ķ是一个正整数。该放射ķ-G 是最小的整数 s 为此功能 FVG{01个2s} 这样对于任何两个顶点 üvVG|Fü-Fv|ķ+1个-düv,在哪里 düv 是之间的距离 üv。特别是当d 是的直径 G, 收音机-d-number被称为无线电号码G。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ķ-树木数量。我们称一棵树,其无线电ķ-number等于此下限a ķ-下界树。其次,我们建立ķ-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ķ 这样一棵给定的树 Ť 是一个 ķ-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ķ下界树的三种方式。最后,我们提出一些问题,以供将来研究。

更新日期:2020-08-27
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