ISRN Combinatorics

Volume 2013 (2013), Article ID 326038, 5 pages

http://dx.doi.org/10.1155/2013/326038

Research Article

Radio Number for Total Graph of Paths

S. K. Vaidya¹ and D. D. Bantva²

¹Department of Mathematics, Saurashtra University, Gujarat, Rajkot 360 005, India

²Department of Mathematics, L. E. College, Gujarat, Morvi 363 642, India

Received 31 August 2012; Accepted 27 September 2012

Academic Editors: A. P. Godbole and D. S. Kim

Copyright © 2013 S. K. Vaidya and D. D. Bantva. 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.

A radio labeling of a graph is a function from the vertex set to the set of nonnegative integers such that , where and are diameter and distance between and in graph , respectively. The radio number of is the smallest number such that has radio labeling with . We investigate radio number for total graph of paths.

1. Introduction

In a telecommunication system to design radio networks, the interference constraints between a pair of transmitters play a vital role. For the transmitters of radio network, we seek to assign channels such that the network fulfills all the interference constraints. The assignment of channels to the transmitters is popularly known as channel assignment problem which was introduced by Hale [1]. For radio network if we assume that the frequencies are uniformly distributed in the spectrum then the frequency span determines the bandwidth allocated for the assignment. In this case, the interference between two transmitters is closely related with the geographic location of the transmitters. Earlier designer of radio networks considered only the two-level interference, namely, major and minor. They classified a pair of transmitters as very close transmitters if the interference level between them is major and close transmitters if the interference level between them is minor.

To solve the channel assignment problem, the interference graph is developed and assignment of channels converted into graph labeling (a graph labeling is an assignment of label to each vertex according to certain rule). In interference graph, the transmitters are represented by the vertices, and two vertices are joined by an edge if corresponding transmitters have the major interference while two transmitters have minor interference then corresponding vertices are at distance two, and there is no interference between transmitters if they are at distance three or beyond it. In other words, very close transmitters are represented by adjacent vertices, and close transmitters are represented by the vertices which are at distance two apart. In fact, Roberts [2] proposed that a pair of transmitters which has minor interference must receive different channels and a pair of transmitters which has major interference must receive channels that are at least two apart. Motivated through this problem Griggs and Yeh [3] introduced -labeling in which channels are related with the nonnegative integers.

Definition 1. A distance two labeling (or -labeling) of a graph is a function from vertex set to the set of nonnegative integers such that the following conditions are satisfied:

(1) if ,(2) if .

The span of is defined as . The -number for a graph , denoted by , is the minimum span of a distance two labeling for . The -labeling is explored in past two decades by many researchers like Yeh [4], Sakai [5], Chang and Kuo [6], Vaidya et al. [7], and Vaidya and Bantva [8].

But as time passed, practically it has been observed that the interference among transmitters might go beyond two levels. Radio labeling extends the number of interference level considered in —labeling from two to the largest possible—the diameter of . The diameter of denoted by diam or simply by is the maximum distance among all pairs of vertices in . Motivated through the problem of channel assignment of FM radio stations, Chartrand et al. [9] introduced the concept of radio labeling of graph as follows.

Definition 2. A radio labeling of is an assignment of positive integers to the vertices of satisfying

The radio number denoted by is the minimum span of a radio labeling for . Note that when diam is two then radio labeling and distance two labeling are identical.

Investigating the radio number of a graph is an interesting and challenging task as well. So far the radio number is known only for handful of graph families. Liu and Zhu [10] have given the radio labeling for paths and cycles. Liu and Xie [11, 12] also studied the case of radio labeling for square of paths and cycles while Der-Fen Liu [13] has given a lower bound for radio number of trees and presented a class of trees achieving the lower bound.

Notice that the expansion of radio network according to certain rule is equivalent to saying that the expansion of interference graph by means of specific graph operation. The expansion of existing network and to determine the radio number for the expanded network is also an interesting task. At the same time, it is also important to relate the radio number of existing network with the expanded network. In this paper, we take up the issue of expansion of linear network in the context of total graph of path and also investigate the radio number for the same.

Definition 3. The total graph of a graph is the graph whose vertex set is and two vertices are adjacent whenever they are either adjacent or incident in . The total graph of is denoted by .

From the definition of total graph, it is clear that the diameter of is same as diameter of , and the center of graph is if and if . Terms not defined here are used in the sense of West [14].

2. Main Results

The radio number of path (linear transmitter network) is investigated by Liu and Zhu [10] as stated in the following result.

Theorem 4. For any ,

Now we focus upon the radio number of the linear network which is expanded by means of total graph operation on . Throughout this work, we denote a path with vertices by , where and .

For the path , let and be the centers. Let be the vertices on left side, and are the vertices on right side with respect to the centers. The edges are . Then the vertex set of total graph of is . For the consistency in notation, we rename the vertices , , , , by , , , , , respectively.

For the path , let be the center. Let , be the vertices on left side, and , are the vertices on right side with respect to the center. The edges are . Then the vertex set of total graph of is . For the consistency in notation, we rename the vertices , , , by , , , , respectively.

Let for , , where

Let for , , where

For the graph , we say two vertices and are on opposite side if or and or .

Define the level function , where is the set of whole numbers with respect to a center vertex by

In graph , the maximum level is if and if .

Observation 1. For ,

(1)(2)

Theorem 5. Let be an assignment of distinct nonnegative integers to , where and be the ordering of such that defined by and . If and , for any then is a radio labeling.

Proof. Let be an assignment of distinct nonnegative integers to such that , , for any and with for any holds, where .

Now we want to prove that is a radio labeling. That is, for any , .

For each , let . Let then

If , then .

Subcase 1 (if is even). Consider

Subcase 2 (if is odd). Consider

Thus, in both the subcases is a radio labeling and hence the result.

Theorem 6. Let be a total graph of path on vertices and . Then if .

Proof. Let be an optimal radio labeling for , where . Then , for all . Summing these inequality, we get

For , we have

Substituting (10) in (9), we get

For , , and

Thus, if .

Theorem 7. Let be a total graph of path on vertices and . Then if .

Proof. For , define by and for all as per following ordering of vertices:

Thus, it is possible to assign labeling to the vertices of with span equal to the lower bound satisfying the condition of Theorem 5 and hence is a radio labeling.

Theorem 8. Let be a total graph of path on vertices and . Then if .

Proof. The proof follows from Theorems 6 and 7.

Theorem 9. Let be a total graph of path on vertices and . Then if .

Proof. Let be an optimal radio labeling for , where . Then − , for all . Summing these inequality, we get

For , we have

Substituting (15) in (14), we get

For , , and ,

Thus, if .

Theorem 10. Let be a total graph of path on vertices and . Then if .

Proof. For , define by and for all as per following ordering of vertices:

Thus, it is possible to assign labeling to the vertices of with span two more than the lower bound which is a radio labeling and hence if .

Example 11. In Figure 1, the ordering of the vertices and optimal radio labeling for is shown.

Figure 1: .

Example 12. In Figure 2, the ordering of the vertices and optimal radio labeling for is shown.

Figure 2: .

3. Open Problem

In connection with Theorems 9 and 10, we feel that it is not possible to find the radio number with span equal to the lower bound but radio labeling for exists with span two more than the lower bound. We strongly believe that in case of , the span will exceed than the lower bound. This feeling gives rise to an interesting problem to investigate exact radio number for , we pose the following conjecture.

Conjecture 13. Consider

4. Concluding Remarks

The establishment of radio transmitters network which is free of interference is the demand of the current time. It has also posed some new challenges. We take up this problem in the context of total graph of paths. We completely determine the radio number of total graph of path . The derived results are applicable for the expansion of linear network of transmitters.

Acknowledgment

The authors are highly thankful to the anonymous referees for their critical comments and kind suggestions on the first draft of this paper.

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