Radio Numbers of Certain -Distant Trees

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