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J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

Radio Number of Uniform Theta Graphs BHARATI RAJAN and KINS YENOKE Department of Mathematics, Loyola College, Chennai 600 034, India ABSTRACT A radio labeling of a connected graph G is an injection f from the vertices of G to the non- negative integers , |

| 1 diam(G) for every two distinct vertices u and w of G. The radio number of f denoted is the maximum number assigned to any vertex of G. The radio number of G denoted is the minimum value of taken over all labeling f of G. In this paper we determine bounds of the radio number of uniform theta graphs. Keywords: labeling, radio labeling, radio number and uniform theta graphs.

1. INTRODUCTION

2. AN OVERVIEW OF THE PAPER

Interest in graph labeling problems began in the mid-1960’s with a conjecture of Ringel20 and a paper by Rosa21. In the intervening years dozens of graph labelings techniques have been studied in over several papers. Despite the large number of papers, there are relatively few general results or methods on graph labeling. Indeed most of the results focus on particular classes of graphs and utilize ad hoc methods. Frequently, the same classes have been done by several authors. Labeled graphs serve as useful models for a broad range of applications such as coding theory, x-ray, crystallography, radar, astronomy, circuit design, channel assignments of FM radio stations and communication network addressing2, 4.

Graph theory models for radio frequency assignment problems can be traced back to the early 1980's in the paper of Hale11. In 2001 Chartrand et al.6 were motivated by regulations for channel assignments of FM radio stations to introduce radio labeling of graphs. Channels assigned to FM radio stations depend not only on the effective radiated power of their signals and the heights of their antennas but also on their distances from other stations. In wireless networks, an important task is the management of the radio spectrum that is the assignment of radio frequencies to transmitters in a way that avoid interferences. Interferences can occur if transmitters with close locations receive close frequencies.

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

In particular, two stations that share the same channel must be separated by at least 115 kilometers; however, the required separation depends on the classes of the two stations. Two channels are first-adjacent, or simply adjacent, if their frequencies differ by 0.2 MHz, that is, if they are consecutive on the FM radio dial. For example, FM stations on channels 98.4 MHz and 98.6 MHz are adjacent. The distance between two radio stations on adjacent channels must be at least 72 kilometers; however, the actual required separation depends on the classes of the two stations. The distance between two radio stations whose channels differ by 400 or 600 kHz (second- or third-adjacent channels) must be at least 31 kilometers. Once again, the actual required separation depends on the classes of the stations. The problem, often modeled as a coloring problem on the graph where vertices represent transmitters and edges indicate closeness of the transmitters. Let G be a connected graph and let k be an integer ≼ 1. The distance between two vertices u and v of G is denoted by d (u, v) and the diameter of G is denoted by diam(G). A radio k-labeling f of G is an assignment of non-negative integers to the vertices of G such that , | | 1 for any two distinct vertices u and v of G. The span of the function f, denoted by ௞ , ismax , . A radio k-labeling with k = diam(G) is known as a radio labeling (or multilevel distance labeling in16). The minimum span of a radio labeling is called the radio number, denoted by .

3. UPPER BOUNDS FOR THE RADIO NUMBER OF UNIFORM THETA GRAPHS A generalized theta graph â‚ , â‚‚ . . . ŕŽ‘ consists of a pair of end vertices joined by Îą internally disjoint paths of lengths â‚ , â‚‚ . . . ŕŽ‘ 1. These end vertices are called north pole (N) and south pole (S) and the paths are called longitudes. A generalized theta graph in which all the paths are of same length is called a uniform theta graph and is denoted by , " [BrInAmPa2005]. Here n denotes the number of internal vertices in each longitude.

Let #â‚ , #â‚‚ . . . #ŕŽ‘ denote the longitudes in the uniform theta graph , " . The number of vertices and edges in , " are " 2 and " 1 respectively. Its diameter is 1. In this paper we obtain bounds for the radio number of uniform theta graphs. We use a specific naming of vertices of the theta graph in order to define a suitable labeling. First let n be odd. We divide the vertex set into two parts called the upper and lower halves. The vertices v's stand for the upper half vertices and the vertices w's for the lower half vertices. The vertices in the upper half are named from left to right starting from the middle of the longitudes and enumerated towards the north pole. Similarly the vertices in the lower half are named from left to right starting from the middle and enumerated towards the south pole. More precisely, let áˆşŕŻ?ିଵáˆťŕ°ˆାŕŻœ % #ŕŻœ , 1 &

' & ", be at distance (ଶ ) 1 *, 1 & * & ௡

(ଶ ), from the north pole N. That is ௡

â‚ , â‚‚ . . . ŕŽ‘ are at distance (ଶ ) from the ௥

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

north pole N arranged left to right; ௥ , ŕŽ‘ାଵ , ŕŽ‘ାଶ . . . ଶŕŽ‘ are at distance (ଶ ) 1 from N arranged left to right etc; finally a set of Îą vertices at distance 1 from N. Let ଵ + , ଵ , ଶ ‌ á‰’ŕł™á‰“ŕŽ‘ .. ŕ°Ž

Similarly let /áˆşŕŻ?ିଵáˆťŕ°ˆାŕŻœ % #ŕŻœ , 1 & ' & ", be at distance

௡ (ଶ ) 1 *,

1&*&

௡ 0ଶ 1,

from

the south pole S. Let ଶ + ,/ଵ , /ଶ ‌ /á‰”ŕł™á‰•ŕŽ‘ ..

Then the vertex set of , " is + â‚ 2 â‚‚ 2 3, 4 . ŕ°Ž

Theorem 3.1: Let G be , " , n odd. Then the radio number of G satisfies & ௡ ଶ

௡ ଶ

5(ଶ ) 0ଶ 1 2 0ଶ 16 " 1, " 7 . ௡

Proof: Define a mapping : 9 3 as follows:

3 + 1,

4 + 2,

áˆşŕŻ?ିଵáˆťŕ°ˆାŕŻœ + ( ) 2* 1 ' * 2 1 ²" * 3, ' + 1,2 . . . ", * + 1,2 . . . ( ) , 2 ଶ

/áˆşŕŻ?ିଵáˆťŕ°ˆାŕŻœ + ( ) " 0 1 2* 1 ' 2 2 *² 1 " * 1, ' + 1,2 . . . ", * + 1,2 . . . 0 1. 2 Now we claim that f is a valid radio labeling. Case 1: , / % â‚ Case 1.1: If u and w are vertices in the same /+ longitude, then + áˆşŕŻ&#x;ିଵáˆťŕŽ‘ା୩ , áˆşŕŻŚŕŹżŕŹľáˆťŕŽ‘ା୩ , 1 & # < & (ଶ ), 1 & & " and ௥

, / 1. Therefore + (ଶ ) 2# ௡

1 # 1 ²" # 3 and / + (ଶ ) 2 1 1 ²" 3. Hence , / | / | 1 |2 # # 1 ² 1 ² "| 2. ௡

Case 1.2: Suppose and / are vertices in different longitudes, then + áˆşŕŻ&#x;ିଵáˆťŕ°ˆା௞ and

/ + áˆşŕŻŚŕŹżŕŹľáˆťŕ°ˆା௠, 1 & # & & (ଶ ) , 1 & < = & ". We identify that the distance between two vertices in same position of different longitudes is exactly the difference between the number of vertices in each longitude increased by 3 and twice the position of the of the vertices from the middle towards the north pole. That is, if p is the position of the vertices, then , / + 3 2>. Also we identify that the distance between two vertices in different positions of different longitudes is atleast the difference between the number of vertices (n) increased by 2 and twice the position of a vertex (p) whose distance from the north pole is maximum. That is , / + 2 2>. ௥

Case 1.2.1: If # < , then ଶ 2 ௡

1 1 ² 3,

2 ௥

1 1 ² 3 and , / ଶ

2# 2. Therefore | | |2 2

1 ² 1 ² |. Here | | is minimum for # + = +

1, + " and + 2. Hence , / | / | 2 2 |2" 4 " 1 1 2 "| + |2" 2 2"| + 2.

Case 1.2.2: If # + , then + @ /2 B 2# 1 # 1 ²" # 3,

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

2 1 1 ² 3 ௡ ଶ

and , / + 2# 3. Therefore

, | | 2 3 ௥ ௥ | 2 1 1 ² 3

2 1 1 ² 3 | 2 3 | 2 1 | 2 3 2 1 2. ଶ

ଶ

Case 2: , / % â‚‚

Case 2.1: If u and w are vertices in the same longitude, then + /áˆşŕŻ&#x;ିଵáˆťŕ°ˆା௞ and

áˆşŕ­ąŕŹżŕŹľáˆťŕŽ‘ା୩ ,1 , 1 . ௥ ଶ

Therefore

/2 "² # /2 $ 2 1 ² 1 1,

/2 "² # /2 $ 2 1 ² 1 1 and , 1.

Hence

, | | 1 |2 ² ² | 2.

Case 2.2: Suppose u and w are vertices in different longitudes, then + /áˆşŕ­ŞŕŹżŕŹľáˆťŕŽ‘ା୩ and / + /áˆşŕ­ąŕŹżŕŹľáˆťŕŽ‘ା୫, 1 & #, & 0ଶ 1 , 1 & < = & ". We identify that the distance between two vertices in same position of different longitudes is exactly the difference between the number of vertices in each longitude increased by 1 and twice the position of the of the vertices from the middle towards the south pole. That is, if > is the position of the vertices, then , / + 1 2>. Also we identify that the distance between two vertices in different positions of different longitudes is atleast the difference between the number of ௥

vertices in each longitude increased by 2 and twice the position of a vertex (>) whose distance from the north pole is maximum. That is , / + 2>.

Case 2.2.1: If # < , then

% & % & 2 1 ଶ

௥

௥

ଶ

ଶ

ଶ

1 1, % & % & ଶ ଶ 2 1 ଶ 1 1 ௡

ଶ

௥

and , / 2#. Therefore

| | ' 2 1 2 1 ( ଶ 1 ଶ 1 ) '.

Here | / | is minimum for # + = + 1, + " and + 2. Hence , | | 2

|3 5 3 1 2| 2 4 2.

Case 2.2.2: If # + , then ଶ

௥

ଶ

௥

+ (Cଶ D) " 0Cଶ D1 2# 1 #² 1 " # 1, ௡

/ + (Cଶ D) " 0Cଶ D1 2# 1 = #² 1 " # 1 and , / + 2# 1. Therefore ௡

, | | 2 1

| % & % & 2 1 ² 1 ௥

ଶ

௥

ଶ

ଶ ଶ

1 % & % & 2 1 ² ଶ ଶ 1 1 | 2 1 | 2 1 | 2. ௡

௥

Case 3: % â‚ and / % â‚‚. This case is possible when 3. Case 3.1: If + áˆźáˆşŕ­ŞŕŹżŕŹľáˆťŕŽ‘ା୩ , and áˆşŕ­ąŕŹżŕŹľáˆťŕŽ‘ା୫ , 1 & & ", 1 & = & " 1,

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

1 & # & (ଶ ) , 1 & & 0ଶ 1, then , / 1 and ௡ ௡ | / | 2 ( ) 0 1 1. ଶ ଶ Therefore d , / | / | 1 ௡ ௡ 2 ( ) 0 1 1 2, since 3. ௡

ଶ

௥

ଶ

Case 3.2: If + á‰’ŕł™á‰“ŕ°ˆ , and / + /ŕŻ&#x;ିଵáˆťŕ°ˆା௞ , ŕ°Ž

1 & & ", 1 & # & 0 1, ଶ ௡

then , / (ଶ ) 2 and ௡

| / |

௡ (ଶ ).

Therefore , / | / + (ଶ ) ௡

2 (ଶ ) + 2. ௡

Case 4: If % â‚ and / + 3 then either ௥ , / 1 and | / | " (ଶ ) or , / + (ଶ ) and | / | 2 ௥

௡ (ଶ ).

Therefore in both the possibilites , / | / | 2.

Case 5: If % â‚ and / + 4 then either , / 1 and | / | " ௥ ௥ (ଶ ) 1 or , / + (ଶ ) and |

/ | 1 (ଶ ). Therefore in both the possibilities , | | 2. ௡

Case 6: If % â‚‚ and / + 3 then ௥ , / 1 and | / | (ଶ ) ²" 0ଶ 1 2. Therefore , | | 2. ௥

Case 7: If , -â‚‚ and / then , 1 ௥ ௥ and | / | (ଶ ) ²" 0ଶ 1 1.

Therefore , | | 2.

Case 8: If + 3 and / + 4 then , / + 1 and | / | + 1. Therefore , / | / | + 2. Hence , / | / | 2 for all , / % and is a valid radio labeling. ௥ ௥ ௥ Thus & C( ) ² 0 1 ² 2 0 1D " 1, ଶ ଶ ଶ " 7 and is odd. â– We next modify the naming of vertices of , " in the case when is even. We divide the vertex set into two equal parts called the upper and lower halves. The vertices v's stand for the upper half vertices and the vertices w's for the lower half vertices. The vertices in the upper half are named from left to right starting ௥ ௧௛

from C D ଶ

vertex of the longitudes and

enumerated towards the north pole. Similarly the vertices in the lower half are named from left to right starting from Cଶ 1D ௡

௧௛

vertex of the longitude and

enumerated towards the south pole. More precisely, let áˆşŕ­¨ŕŹżŕŹľáˆťŕŽ‘ା୧∈#ŕŻœ , 1 & ' & ", be at

distance Cଶ D 1 *, 1 & * & Cଶ D from the ௡

௥

north pole N. That is â‚ , â‚‚ . . . ŕŽ‘ are at distance Cଶ D from the north pole N arranged ௥

left to right; áˆşŕ°ˆାଵáˆť , áˆşŕ°ˆାଶáˆť ‌ ଶŕ°ˆ are at

distance Cଶ D-1 from N arranged left to right ௡

etc; finally a set of " vertices at distance 1 from N. Let â‚ + E áˆşŕ°ˆାଵáˆť , áˆşŕ°ˆାଶáˆť ‌ ଶŕ°ˆ F and ₃ + â‚ , â‚‚ . . . ŕ°ˆ .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

Theorem 3.2: Let be , " , even. Then the radio number of satisfies & IC ଶ D 2J " ଶ 3, " 7 . ௡మ

௥

Proof: Define a mapping : 9 3 as follows:

3 + 1, 4 + 2.

" 2 ( ) 2' 1, ' 2 2 " + 1,2 . . . 0 1 . 2 "

ଶŕŻœିଵ + 2' 1, ' + 1,2 . . . ( ). 2 2 "

/ଶŕŻœ + 2 ' 1 , ' + 1,2 . . . 0 1. 2 2

ଶŕŻœ +

" " 2 ( ) 2', ' + 1,2 ‌ ( ). 2 2 2

K ŕ°ˆŕŻ?ାŕŻœ L + 2' * 1 * ଶ * " * 2 1, ' + 1,2 . . . ", * + 1,2 . .. 1. 2 ଶ

K/ŕ°ˆŕŻ?ାŕŻœ L + I J C D " 2' * 1 4 2 * * 1 2 " * 2, ' + 1,2 . . . ", * + 1,2 . .. 1. 2

/ଶŕŻœିଵ +

Figure 3.1: A uniform theta graph G with , and a radio labeling realizing the upper bound

Similarly let áˆşŕ­¨ŕŹżŕŹľáˆťŕŽ‘ାŕŻœ , ŕŻœ , 1 0 ,

be at distance Cଶ D 1 *, 1 & * & Cଶ D, from the south pole S. ௡

௥

Let â‚‚ + ,/ŕ°ˆାଵ , /ŕ°ˆାଶ . . . /á‰€ŕł™á‰ ŕ°ˆ . ŕ°Ž

and â‚„ + /â‚ , /â‚‚ . . . /ŕ°ˆ . Then the vertex set of , " is + â‚ 2 â‚‚ 2 ₃ 2 â‚„ 2 3, 4 .

Proceeding as in Theorem 3.1 we can show that f is a radio labeling and that ଶ 12 3 24 3, 2 2 5 .

Figure 3.2 shows the uniform theta graph 4, 5 and a radio labeling. â–

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

880

REFERENCES 1.

2.

3.

4.

5. Figure 3.2: A uniform theta graph G with , and a radio labeling realizing the upper bound.

4. CONCLUSION In this paper we have determined the bounds for the radio number of uniform theta graphs. The same problem for honeycomb mesh, Silicate network, oxide network and Torus mesh are under investigation.

6.

7.

8.

Bharati Rajan, Indra Rajasingh, Kins Yenoke, Paul Manuel, “Radio number of graphs withsmall diameter”, International Journal of Mathematics and Computer Science, Vol.2, pages 209-220, (2007). Bharati Rajan, Indra Rajasingh, KinsYenoke,“Radio number of wheel related graphs”, Proceedings of the international conference of Mathematics and Computer Science, Vol.1, pages 178-182, (2009). Bharati Rajan, Indra Rajasingh, Amutha A and Paul Manuel, "Minimum Tree Spanner Problem for Generalized Theta Graphs", Third National Conference Mathematical and undirected graphs”, Proc. IEEE, Vol. 65, pages 562-570, (1977). G. S. Bloom and S.W. Golomb, “Application of undirected graphs”, Proc. IEEE, Vol. 65, pages 562-570, (1977). G. S. Bloom, and S. W. Golomb, “Numbered complete graphs, unusual rulers, and assorted applications” Theory and applications of graphs, Lecture notes in Mathematics 642. Springer Verlag, Newyork, pages 5365, (1978). G. Chartrand, D. Erwin, P. Zhang, “A radiolabeling problem suggested by FM channel restrictions”, preprint. G. Chartrand, L. Nebesk´y and P. Zhang.Radio k-colorings of paths. Discussiones Mathematicae Graph Theory, 24:5-21, (2004). G. Chartrand, D. Erwin, and P. Zhang. Radio antipodal colorings of cycles.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

881

9.

10.

11.

12.

13.

14.

15.

Bharati Rajan, et al., J. Comp. & Math. Sci. Vol.2 (6), 874-881 (2011)

Congressus Numerantium, 144:129– 141, (2000). G. Chartrand, D. Erwin, and P. Zhang. Radio antipodal colorings of graphs. Math. Bohemica. J. R. Griggs and R.K. Yeh. Labelling graphs with a condition at distance 2. SIAM J. Disc. Math., 5:586–595, (1992). W. K. Hale: Frequency assignment: Theory and applications, Proceedings of the IEEE 68, (1980). M. Kchikech, R. Khennoufa and O. Togni.Radio k-labelings for Cartesian products of graphs. Discussiones Mathematicae Graph Theory, to be published. M. Kchikech, R. Khennoufa and O. TogniLinear and cyclic radio klabelings of trees. Discussiones Mathematicae Graph Theory, 27 (1): 105-123, (2007). G. Chartrand, D. Erwin, P. Zhang and F. Harary, “Radio labelings of graphs”, Bull. Inst. Combin. Appl., Vol 33, pages 77-85, (2001). D. Kuo and J.-H. Yan. On L(2, 1)labelings of Cartesian products of path and cycles Discrete Math., 283(1-3): 137-144, (2004).

16. Liu and X. Zhu, Multi-level distance labeling for paths and cycles, preprint. 17. D. Liu and M. Xie. Radio Number for SquarePaths. Ars Combinatoria,to appear. 18. D. Liu and M. Xie. Radio Number for Square Cycles. Congr. Numerantium, 169 : 105 - 125, (2004). 19. D. Liu. Radio number for trees. Manuscript, (2006). 20. G. Ringel, “Theory of Graphs and its applications”, Proceedings of the Symposium Smolenice 1963, Prague Publ. House of Czechoslovak Academy of Science, pages162, (1964). 21. A. Rosa, “Cyclic Steiner triple systems and labelings of triangular cacti”, Scientia, Vol. 1, pages 87-95, (1988). 22. Wong G.K., Coppersmith D.A.; “A combinatorial Problem related to Multitude Memory Organization”, Journal of Association for Computing Machinery, Vol. 21, 392-401, (1974). 23. J. Xu, “Topological structure and analysis of interconnection Networks”, Kluwer Academic Publishers, ISBN 14020-0020-0, (2001). 24. P. Zhang, “Radio labelings of cycles”, Ars combin., Vol. 65, pages 21-32, (2002).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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