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 + + .
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 of , the span of , where ’s are arranged as in Definition 1.
Proof. Consider = .
Since , we get .
Lemma 3 says that to obtain a minimal radio coloring of a graph, one should maximize and minimize 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.
Theorem 4. If is a radio coloring of such that and : is a radio coloring of , then is a minimal radio coloring of .
Proof. For any radio coloring of , Lemma 3 gives that . Then
Now, we determine the radio number of an -distant tree with , , , and the degrees of the vertices on the central path satisfy certain conditions (given in the theorem below).
Theorem 5. Let be an -distant tree of order with a central path : , and satisfy (i);(ii), for , ;(iii), , ;(iv)for every , , the number of vertices at distance and lying on a branch incident on , , is a constant say ().
Then .
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 .
Algorithm 6
Step I. We make an ordering of the vertices on the central path as , , , , , , , , , that is, , .
Step II. Let be equal to the vertices , , , , , respectively. Let be an ordered tuple of vertices in such that is at distance from and lies on a branch incident on , , and . So for any , there are disjoint such tuples, . Consider the sequence : , where .
Step III. We take . In this case, there are disjoint tuples . Select an arbitrary such tuple and use the first terms of the sequence to name the vertices in in order. For the next tuple we use the next 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 from , in the similar manner. We proceed like this until we name all the vertices in and are of distance from .
Step V. Consider the sequence . The terms in the beginning of are assigned (or named) to the distance one vertices in adjacent to and alternately (the number of distance one vertices in adjacent to and , are equal). The next terms of are assigned to the distance one vertices in adjacent to and alternately, and so on, till we name all the vertices of . Observe that .
Now is an ordering of all the vertices of .
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 .
Example 7. In this example, we illustrate the above coloring by considering the -distant tree given in Figure 2
Here , the central path : , , , , , , , , , , , , , , , , , , , , , , , , , : , and : . So the ordering of the vertices of this -distant tree is in Figure 3.
The assignment is given in Figure 4.
Continuation of the Proof of Theorem. We first show that is a radio coloring. We need to check that , (we call this as radio condition). From the definition of , radio condition holds true for pair of vertices and , . For ,
Since
radio condition holds true for all the pair of vertices and , where one of them is on the central path and the other in .
Now,
Since are at least distance apart from and have colors greater than , radio condition automatically holds true, and
If is at a distance from , then
Since , radio condition holds true between and all other vertices.
Let . Then is the last vertex at distance from . Since, from the definition of , radio condition holds true for and , and , , radio condition automatically holds true between and , .
Since , , and , we have
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
So, each term in the distance sum of a radio coloring contains two indices from with different signs because from (9), distance between every pair of vertices contains , . Since there are terms in the distance sum, it contains elements from the set with half positive and half negative sign. Also, the indices and occur at most twice, and the index occur at most times, where . So, the possible maximum distance sum is + + + + .
Now, the distance sum
where coincides with the possible maximum distance sum above and is equal to .
One observes that . Therefore, from Theorem 4, the radio coloring is minimal and
As a consequence of the above theorem, we determine the radio number of a class of caterpillars (-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 apart have the same degree (where is the total number of vertices on the central path).
Corollary 8. Let be a caterpillar of order and with a central path . If , , , then
Proof. This is 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.
Here , the central path : and . So the ordering of vertices of the caterpillar is illustrated in Figure 6 and the coloring is given in Figure 7.
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.
Corollary 10. Let be a caterpillar of order with a central path : . If , , , then .
Proof. One can prove this result by substituting , in Corollary 8.
Example 11. In this example we illustrate Corollary 10 by considering the caterpillar given in Figure 8.
Here , the central path : and . So the ordering of vertices of the caterpillar is illustrated in Figure 9 and the coloring is given in Figure 10.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are thankful to the referee for his/her valuable comments and suggestions which improved the presentation of the paper.