J. Comp. & Math. Sci. Vol.2 (6), 868-872 (2011)
Radio Antipodal Number of Certain Graphs ALBERT WILLIAM and CHARLES ROBERT KENNETH Department of Mathematics, Loyola College, Chennai, India ABSTRACT Let , be a graph with vertex set and edge set . Let denote the diameter of and , denote the distance between the vertices and in . An antipodal labeling of with diameter is a function that assigns to each vertex , a positive integer , such that , | | , for all , . The span of an antipodal labeling is | |: , . The antipodal number for , denoted by , is the minimum span of all antipodal labelings of . Determining the antipodal number of a graph G is an NPcomplete problem. In this paper we determine the antipodal number of certain graphs. Keywords: Labeling, radio antipodal numbering, diameter.
1. INTRODUCTION Let G be a connected graph and let be an integer, 1. A radio -labeling of is an assignment of positive integers to the vertices of such that , | | 1 for every two distinct vertices and of , where , is the distance between any two vertices and of . The span of such a function , denoted by | |: , . Radio -labeling was motivated by the frequency assignment problem4. The maximum distance among all pairs of vetices in G is the diameter of G. The radio labeling is a radio -labeling when . When 1, a radio -labeling is called a radio antipodal
labeling. In otherwords, an antipodal labeling for a graph G is a function, : 0,1,2, ‌ such that , | | . The radio antipodal number for G, denoted by ! , is the minimum span of an antipodal labeling admitted by G. A radio labeling is a one-to-one function, while in an antipodal labeling, two vertices of distance
apart may receive the same label. The antipodal labeling for graphs was first studied by Chartrand et al.7, in which, among other results, general bounds of ! were obtained. Khennoufa and Togni9 determined the exact value of ! "
for paths " . The antipodal labeling for cycles # was studied in5, in which lower
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
869
Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 868-872 (2011)
bounds for ! # are obtained. In addition, the bound for the case ! $ 2 % 4 was proved to be the exact value of ! # , and the bound for the case ! $ 1 % 4 was conjectured to be the exact value as well6. Justie Su-tzu Juan and Daphne Der-Fen Liu8 confirmed the conjecture mentioned above. Moreover they determined the value of ! # for the case ! $ 3 % 4 and also for the case ! $ 0 % 4 . They improve the known lower bound5 and give an upper bound. They also conjecture that the upper bound is sharp. In this paper we obtain an upper bound for the radio antipodal number of the hexagonal mesh and grid. 2. THE RADIO ANTIPODAL NUMBER OF HEXAGONAL MESH We denote the hexagonal mesh of dimension n by () . Let *, + and , be the
three axis inclined mutually at an angle 60 respectively and let + , + ,… + be the β lines (vertical lines) marked from left to right as shown in the figure. We name vertices on + , + ,… + lines as follows: We name the vertices on + from top to bottom as , , … , the vertices on + from top to bottom , , … . Finally we name the vertices on + from top to bottom as మ , మ , … మ . The diameter of () is 2n-2. We first provide an upper bound for the radio antipodal number of the hexagonal mesh of dimension n.
v28 v19 v12 v6 v1 v2 v3
v9
v4 v10 v5
v22
v15
v39 v31 v32
v23 v16 v24
v33
v47 v40
v26 v18 v27
v41
v53
v58
v54
v59
v55
v60
v56
v61
v57
v62
v49 v42 v50 v43 v51
v35 v36
v52
v48
v34
v17 v11
v45 v46
v30 v21
v14
v37 v38
v20
v13
v7 v8
v29
v44
Figure 1(a)
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 868-872 (2011)
870
183 4
127 190
78 134
36
141
43 8
148 211
29
162
64
169 176
130
165
137
172
95 144
46 232
120
158
88 39
225
113 71
123 81
32 218
106 22
151
74 25
155
57
116
18
99 15
67
204
92 50
109
197
85
1
60 11
179
102 53
239
Figure 1(b): Radio antipodal number of ૾ with diameter 8
Theorem 1 The radio antipodal number of () satisfies
2 3 3 1 1! 1. 2
Proof. Define a mapping : V () . 2! 3 1 1, ! 1,2, ‌ 3! 1
2 / 3! 1 0 1 2 2! 3
ŕ°Ž
1 1, 1,2, ‌ We claim that , | | 2! 2 for all , ()
Case(i): , + , 1 3 3 !.Let and , 1 3 4, 3 3! 1 , 4 5 . Then , 1 and
, | | 1 | 2 3 # 1 2 3 $ 1 | 2 2.
Case(ii): + , 6+ 7, 1 3 , 8 3 !, 5 8. Let and
, 1 3 4, 3 3! 1 , 4 5 .
Then , 1 and
, | | 1 | 2 3 # 1 2 3 $ 1 | 2 2.
Case(iii): , + , ! 1 3 3 2! 1.Let 1 2 2! 3 4 1 and 1 2 2! 3 1 , 1 3 4, 3
3!
5! 2 , 4 5 .
Then , 1 and
, | | 1 | 2 3 # 1 2 3 $ 1 | = 1+%& ' 2 3 #
1 & ' 2 3 $ 1 % 2 2.
Case(iv): + , 6+ 7, ! 1 3 , 8 3 2! 1. Let ŕł™ and ŕ°Ž
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
871
Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 868-872 (2011)
ŕł™ , 1 3 4, 3 3! 5!
ŕ°Ž
2 , 4 5 .
Then , 1 and
, | | 1 | 2 3 # 1 2 3 $ 1 | =1+%& ' 2 3 #
1 & ' 2 3 $ 1 % 2 2.
3. THE RADIO ANTIPODAL NUMBER OF GRID An n-dimensional mesh : , , ‌ has , , ‌ : 1 3 3 , 1 3 3 ! for its vertex set and vertices ‌ ‌ and ‌ ‌ , 1 3 3 ! are adjacent in : , , ‌ . A mesh is a bipartite graph. If at least one side has even length, the mesh has a hamiltonian cycle. A hamiltonian path exists always. Meshes are not regular, but the degree of any vertex is bounded by 2!. Of course, the degree of a corner vertex is less than the degree of an internal vertex. See figure 2(a).
1
78
155
232
19
96
173
12
89
166
243
30
107
184
23
100
177
254
41
118
195
34
111
188
265
52
129
206
45
122
199
276
63
140
217
56
133
210
287
74
151
228
67
144
221
298
85
162
239
Figure 2(b): Radio antipodal number of áˆşŕ˘”ŕľˆŕ˘”áˆť with diameter 12
Theorem 2: The radio antipodal number of satisfies áˆşŕŻĄŕľˆŕŻĄáˆť 1 1 1 1. 2
Proof. Define a mapping : V( ) * +.
( ) 1 ( , 1 1 ) 1, 1,2, ‌ , , 1,2, ‌ . / 2 0 , 1 2 2 3 & '! 2 1 ( , 1 1 ), 1,2, ‌ ,
v15 v22
v29
v36 v43
v1
v8
v2
v9 v16 v23 v30 v37 v44
v3 v10 v17 v24 v4
v31 v38 v45
v11 v18 v25 v32 v39 v46
v5 v12 v19 v26 v33 v40 v47 v6
v13 v20 v27 v34 v41 v48
v7 v14
v21 v28 v35 v42 Figure 2(a)
v49
, 1,2, ‌ & '. It is easy to verify that
, | | 2 1 for all
, ( ). See figure 2(b).
4. CONCLUSION The study of radio antipodal number of graphs has gained momentum in recent years. Very few graphs have been proved to have radio antipodal labeling that attains the radio antipodal number. In this paper we have determined the bounds of the radio antipodal number of the hexagonal mesh and grid. Further study is taken up for various other classes of graphs.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 868-872 (2011)
REFERENCES
of Graphs”, (2000). Chartrand G,Erwin D, and Zhang P, “Radio Labeling of Graphs”, Bull. Inst. Combin. Appl, 33, 77-85 (2001). 8. Justie Su-tzu Juan and Daphne Der-Fen Liu,, “Antipodal Labeling for Cycles”, (2006). 9. Khennoufa R and Tongni O, “A note on Radio Antipodal Colouring of Paths”, Math. Bohem.130 (2005). 10. Mustapha Kchikech, Riadh Khennoufa and Olivier Tongi, “ Linear and Cyclic Radio k-Labelings of Trees”, Discussiones Mathematicae Graph theory, (2007). 11. Ringel G, “Theory of Graphs and its Applications”, Proceedings of the Symposium Smolenice 1963, Prague Publ.House of Czechoslovak Academy of Science, 162 (1964). 12. Rosa A, “Cyclic Steiner Triple Systems and Labeling of Triangular Cacti”, Scientia Vol 1, 87-95, (1988).
7. 1.
2.
3.
4.
5.
6.
Bharati Rajan, Indra Rajasingh, Kins Yenoke, Paul Manuel, “Radio Number of Graphs with Small Diameter”, International Journal of Mathematics and Computer Science, Vol 2, 209-220 (2007). Calamoneri T and Petreschi R, “L(2,1)Labeling of Planar Graphs”, ACM, 2833 (2001). Chang G.J and Lu C, “Distance - Two Labeling of Graphs”,European Journal of Combinatorics, 24, 53-58 (2003). Chartrand G, Erwin D and Zhang P, “Radio k-Colorings of Paths”, Disscus Math. Graph Theory, 24, 5-21 (2004). Chartrand G, Erwin D, and Zhang P, “Radio Antipodal Colorings of Cycles”, Congressus Numerantium, 144 (2000). Chartand G, Erwin D, Zhang P, Kalamazoo “Radio Antipodal Coloring
872
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)