Radio Number for Total Graph of Paths

A radio labeling of a graph GG is a function ff from the vertex set VVVGGV to the set of nonnegative integers such that |ffVffV f ffVffV| f diamVGGV G G f GGGGVffu ffV, where diamVGGV and GGGGVffu ffV are diameter and distance between ff and ff in graphGG, respectively. e radio number rnVGGV of GG is the smallest number kk such that GG has radio labeling with max{ffVffV f ff f VVVGGVf f kk. We investigate radio number for total graph of paths.


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 ful�lls all the interference constraints.e 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.ey classi�ed 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.De�nition 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 satis�ed: e span of  is de�ned as max{|  |     .e -number for a graph , denoted by , is the minimum span of a distance two labeling for .e  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 .e 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.
���n���on �� A radio labeling  of  is an assignment of positive integers to the vertices of  satisfying   −   ≥ diam  +  −      ∀     . ( e radio number denoted by rn 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 speci�c graph operation.e 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.
���n���on �� e 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 .e total graph of  is denoted by .
From the de�nition of total graph, it is clear that the diameter of    is same as diameter of   , and the center of graph    is   if    +  and  3 if   .Terms not de�ned here are used in the sense of �est [14].

Main Results
e radio number of path (linear transmitter network) is investigated by Liu and Zhu [10] us, in both the subcases  is a radio labeling and hence the result.eorem 6.Let   ) be a total graph of path   on  vertices and  = .en rn  )) ≥ 4  )   if  = .
Example 11.In Figure 1, the ordering of the vertices and optimal radio labeling for  10 ) is shown.

Open Problem
In connection with eorems 9 and 10, we feel that it is not possible to �nd the radio number with span equal to the lower bound but radio labeling for  1 ) exists with span two more than the lower bound.We strongly believe that in case of  1 ), the span will exceed than the lower bound.is feeling gives rise to an interesting problem to investigate exact radio number for  1 ), we pose the following conjecture.

Concluding Remarks
e 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   ).e derived results are applicable for the expansion of linear network of transmitters.

Conjecture 13 .
Considerrn  1  =    .(21) as stated in the following result.  .roughout this work, we denote a path with  vertices by   , where     {      …    } and     {      + ∶     …   − }.For the path   , let   and   be the centers.Let       …   − be the vertices on le side, and   ,    …   − are the vertices on right side with respect to the centers.e edges are                  ,          …   −   −   −       ,             …   −   −   − .en the vertex set of total graph of   is     {  ,          …   −        …   −           …,  −        …   − }.For the consistency in notation, we rename the vertices   ,   ,    …   − ,   , For the path  + , let   be the center.Let   ,    …,   be the vertices on le side, and   ,    …    are the vertices on right side with respect to the center.e edges are                   …      − ,                      …      − ,   .en 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  ′  .De�ne the level function  ∶    → , where  is the set of whole numbers with respect to a center vertex  by       for any        .Let  be an assignment of distinct nonnegative integers to   )), where  =  and {  ,   ,  3 ,…,   } be the ordering of   )) such that   ) <   ) de�ned by   ) = 0 and   ) =   )           ).If      )     and      ) ≠      ), for any        then  is a radio labeling.Proof.Let  be an assignment of distinct nonnegative integers to   )) such that   ) = 0,   ) =   )         ), for any      and      )   with      ) ≠      ) for any        holds, where  = .Now we want to prove that  is a radio labeling.at is, for any  ≠ , |  )    )| ≥          ).For each  =   …    , let   =   )    ).Let  ≥    then          =       ⋯    =      )                  ⋯         .
(5)rem 4. For any  ≥ 3, rn        −  +  if      +  if    + .(2)Nowwefocusupon the radio number of the linear network which is expanded by means of total graph operation on(5)In graph   , the maximum level is  −  if    and  if    + .≥           .