Radio Numbers of Certain m-Distant Trees

Radio coloring of a graph G with diameter d is an assignment f of positive integers to the vertices of G such that |f(u) − f(V)| ≥ 1 + d − d(u, V), where u and V are any two distinct vertices of G and d(u, V) is the distance between u and V. The number max {f(u) : u ∈ V(G)} is called the span of f. The minimum of spans over all radio colorings of G is called radio number of G, denoted by rn(G). Anm-distant tree T is a tree in which there is a path P of maximum length such that every vertex in V(T) \ V(P) is at the most distance m from P. This path P is called a central path. For every tree T, there is an integer m such that T is a m-distant tree. In this paper, we determine the radio number of some m-distant trees for any positive integer m ≥ 2, and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars).


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 , 1 ≤  ≤ , specifically, a radio -coloring of  is an assignment  of positive integers to the vertices of  such that |() − (V)| ≥ 1 +  − (, V), where  and V are any two distinct vertices of  and (, V) is the distance between  and V.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][4][5][6][7][8][9].
Although the positive integer  can have value in-between 1 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  1, as ( − 1) + 2 and have also given an upper bound for radio -chromatic number,   (),  ≥ 2, of an arbitrary non-star-tree  on  vertices as (−1)(−1).Liu [5] has given a lower bound for the radio number () of an -vertex tree with diameter  as ( − 1)( + 1) + 1 − 2(), where () is the weight of  defined as () = min ∈() {∑ V∈() (, V)}.She also has characterized the trees achieving this bound.In the same paper, Liu considered spiders denoted by   1 , 2 , 3 ,...,  , which are trees having a vertex V of degree  ≥ 3, and  number of paths of length  1 ,  2 , . . .,   whose one end vertex is V and other ends are pendant vertices.She has given a lower bound for the radio number of   1 , 2 , 3 ,...,  as and has also characterized the spiders achieving this bound.Li et al. [10] have determined the radio number of complete -ary trees ( ≥ 3) with height  (≥2), denoted by  , , as In this paper, we determine the radio number of some -distant trees for any positive integer  ≥ 2, and as a consequence of it, we find the radio number of a class of 1distant trees (or caterpillars).

Radio Numbers of Some 𝑚-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 For every tree , there is an integer  such that  is a -distant tree.Usually 1-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 V, the quantity |() − (V)| − {1 +  − (, V)} 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  1 ,  2 , . . .,   of vertices of  with (  ) ≤ ( +1 ), 1 ≤  ≤  − 1, we define   (or    to specify the coloring ) = {(  ) − ( −1 )} − {1 +  − (  ,  −1 )}, 2 ≤  ≤ .It is clear from the definition of radio coloring that   ≥ 0, for all .With respect to the ordering of vertices of  induced by , we denote () = ∑  =2 (  ,  −1 ).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.
The following lemma gives the span of a radio coloring of a graph of order  in terms of , , distance sum, and 's sum.

Lemma 3. For any radio coloring 𝑓 of 𝐺, the span of
, where   's are arranged as in Definition 1.

Proof. Consider 𝑓(𝑥
Lemma 3 says that to obtain a minimal radio coloring of a graph, one should maximize () and minimize ∑  =2 over all possible radio colorings of the graph.Since this fact is the basic concept to construct a minimal radio coloring, we express it as the theorem below.
Proof.For any radio coloring  of , Lemma 3 gives that Now, we determine the radio number of an -distant tree  with () = 2 − 1,  ≥ 2,  ≤  − √ − 1, and the degrees of the vertices on the central path satisfy certain conditions (given in the theorem below).
Proof.The idea is to define a radio coloring  of  and show that  is minimal by Theorem 4. We first give an algorithm to order the vertices of .
Step III.We take  = .In this case, there are   disjoint tuples .Select an arbitrary such tuple  and use the first 2 − 2 terms of the sequence  to name the vertices in  in order.For the next tuple  we use the next 2−2 terms of the sequence  to name the vertices in  in order.We proceed like this until we cover all the   disjoint tuples.
Step IV.We name the vertices in  \  which are at distance −1 from , in the similar manner.We proceed like this until we name all the vertices in  \  and are of distance 2 from .
Step V. Let  be a coloring to the vertices of  defined by ( Before we prove that  is a minimal radio coloring of , we give an illustration of .
The assignment  is given in Figure 4.
In a similar manner, one can check the radio condition for the remaining pair of vertices.Therefore,  is a radio coloring.Next, we show that  is minimal.Let  and  be any two vertices of .Then we get vertices   and   on the central path  such that  =   or  is on a branch incident on   , and  =   or  is on a branch incident on   .Then (  ,  −1 ) , (10) where  = 2+∑   As a consequence of the above theorem, we determine the radio number of a class of caterpillars (1-distant trees).In the corollary below, we find radio number of caterpillars of odd diameter in which the degrees of every pair of nonpendant vertices on the central path lying at distance  − 1 apart have the same degree (where 2 is the total number of vertices on the central path).
Corollary 8. Let  be a caterpillar of order  and with a central path  : Proof.This is  = 1 case of Theorem 5.The ordering of vertices in this case includes Step I and Step V of algorithm in the proof of Theorem 5 with only variation that if a vertex on the central path is not adjacent to any pendant vertex, then we move to the next possible vertex.
Example 9.In this example, we illustrate Corollary 8 by considering the caterpillar given in Figure 5.
The corollary below is also a consequence of Theorem 5 in which we find the radio number of caterpillars of odd diameter in which all nonpendant vertices on the central path are of the same degree.Example 11.In this example we illustrate Corollary 10 by considering the caterpillar given in Figure 8.
Here  = 6, the central path :  1  2 ⋅ ⋅ ⋅  12 and  = 2.So the ordering of vertices of the caterpillar is illustrated in Figure 9 and the coloring  is given in Figure 10. a

Figure 6 :
Figure 6: Ordering of vertices of the caterpillar given in Figure 5.
(9)So, each term in the distance sum of a radio coloring contains two indices from {1, 2, ..., 2} with different signs because from(9), distance between every pair of vertices contains Figure 7: Radio coloring  of the caterpillar given in Figure 5.Figure 8: A caterpillar for illustration of Corollary 10.Figure 10: Minimal radio coloring of the caterpillar given in Figure 8.