Journal of Discrete Mathematics

Volume 2014, Article ID 486354, 6 pages

http://dx.doi.org/10.1155/2014/486354

## Radio Numbers of Certain -Distant Trees

^{1}Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Karnataka 575025, India^{2}Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Received 22 August 2014; Revised 30 November 2014; Accepted 3 December 2014; Published 15 December 2014

Academic Editor: Tiziana Calamoneri

Copyright © 2014 Srinivasa Rao Kola and Pratima Panigrahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

*Radio coloring* of a graph with diameter is an assignment of positive integers to the vertices of such that , where and are any two distinct vertices of and is the distance between and . The number max is called the *span* of . The minimum of spans over all radio colorings of is called *radio number* of , denoted by . An *m*-*distant tree T* is a tree in which there is a path of maximum length such that every vertex in is at the most distance from . This path is called a *central path*. For every tree , there is an integer such that is a -distant tree. In this paper, we determine the radio number of some -distant trees for any positive integer , and as a consequence of it, we find the radio number of a class of 1-distant trees (or *caterpillars*).

#### 1. Introduction

The* channel assignment problem* is the problem of assigning frequencies to transmitters in some optimal manner and with no interferences; see Hale [1]. Chartrand et al. [2] introduced* radio **-colorings* of graphs which is a variation of Hale’s channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph subject to certain constraints involving the distance between the vertices. For any simple connected graph with diameter and a positive integer , , specifically, a radio -coloring of is an assignment of positive integers to the vertices of such that , where and are any two distinct vertices of and is the distance between and . The maximum color (positive integer) assigned by to some vertex of is called the* span* of , denoted by . The minimum of spans of all possible radio -colorings of is called the radio *-*chromatic number of , denoted by . A radio -coloring with span is called minimal radio -coloring of . Radio -colorings have been studied by many authors; see [3–9].

Although the positive integer can have value in-between and , the case has become a special interest for many authors. Radio -coloring is simply called radio coloring and radio -chromatic number is radio number. Here we concentrate on radio number of trees. Kchikech et al. [4] have found the exact value of the radio -chromatic number of stars as and have also given an upper bound for radio -chromatic number, , , of an arbitrary non-star-tree on vertices as . Liu [5] has given a lower bound for the radio number of an -vertex tree with diameter as , where is the weight of defined as . She also has characterized the trees achieving this bound. In the same paper, Liu considered spiders denoted by , which are trees having a vertex of degree , and number of paths of length whose one end vertex is and other ends are pendant vertices. She has given a lower bound for the radio number of as , where , and has also characterized the spiders achieving this bound. Li et al. [10] have determined the radio number of complete -ary trees () with height (≥2), denoted by , as .

In this paper, we determine the radio number of some -distant trees for any positive integer , and as a consequence of it, we find the radio number of a class of -distant trees (or* caterpillars*).

#### 2. Radio Numbers of Some -Distant Trees

Recall that an *-*distant tree is a tree in which there is a path of maximum length such that every vertex in is at the most distance from . This path is called a* central path*. Since we consider the path as a path of maximum length, the end vertices of are of degree one vertices in the -distant tree; that is, if : is a central path of , then . For every tree , there is an integer such that is a -distant tree. Usually -distant trees are known as caterpillars.

Before we present the main result of the paper, we give a definition and a lemma below which will be used in the sequel. From the definition of a radio coloring , one observes that for any two vertices and , the quantity is an excess for to achieve the minimum span. In the definition, we give notation for these excesses corresponding to pair of vertices. In the lemma, we show that to get an optimal radio -coloring, one has to minimize this sum of excesses.

*Definition 1. *For any radio coloring of a simple connected graph on vertices and an ordering of vertices of with , , we define (or to specify the coloring ) , . It is clear from the definition of radio coloring that , for all . With respect to the ordering of vertices of induced by , we denote . In other words, every radio coloring is associated with a unique positive integer called* distance sum* of .

*Example 2. *In this example, we explain Definition 1.

In Figure 1, a radio coloring of a tree is given. The labels are an ordering of vertices of with , . Here , , , , , , , , , , , and + + .