Upper Bound for the Radio Number of Some Families of Sunlet Graph

ISSN (Print) : 0974-6846 Indian Journal of Science and Technology, Vol 9(46), DOI: 10.17485/ijst/2016/v9i46/98606, December 2016 ISSN (Online) : 0974-5645 Upper Bound for the Radio Number of Some Families of Sunlet Graph

A.Vimala Rani* and N. Parvathi Department of Mathematics, Faculty of Science and Humanities, SRM University, Kattankulathur-603 203 Tamil Nadu, India; [email protected], [email protected]

Abstract Objectives: In this study, Radio Coloring is used to color the graphs. The objective of this article is to analyze the bounds of Line, Middle, Total and Central graphs of Sunlet graph. Methods/Analysis: Combinatorics is an important part of discrete mathematics that solves counting problems without actually enumerating all possible cases. Combinatorics has wide applications in Computer Science, especially in coding theory, analysis of algorithms and others. An equation that

expresses an, the general term of the sequence {an} is called a recurrence relation. Using the generating function of a sequence and few coloring techniques we prove the results. Findings: or optimal k arises in the concept of radio frequency assignment. The radio chromatic score rs(G)17 of a radio coloring is the number of used colors. The number of colors used in a radio coloring The with Problem the minimum of finding score radio is thecoloring radio withchromatic small rn(G) of G. 10 The radio chromatic number of Sunlet graph Sn is 4 if n=3i, i=1,2,... 5 if n=3i+1, i=1,2,... 6 if n=3i+2, i=1,2,...

and is improved to the radio chromatic number of Sunlet graph Sn is 5 if n is congruent to 1mod 3 4 if otherwise

In this paper we improve the radio chromatic number of Sunlet graph Sn and obtain the radio number of Line, Middle, Total

and Central graphs of Sunlet graph. Radio Coloring has wide range of significance because radio coloring has its applications Applications: in communication theory. The paper contributes the researches in the field of computer science and combinatorics.

problem assuming transmitters as vertices and interference as adjacencies between two vertices. Radio coloring is of great significance because the frequency assignment problem is modeled as a graph coloring Keywords: Sunlet Graph, Line Graph, Middle Graph, Radio Number, Total Graph and Central Graph

1. Introduction Yeh introduced a problem proposed by Roberts which they call the Radio Coloring problem. It is the problem Radio frequency assignment is a broad area of research. of assigning radio frequencies (integers) to transmitters The task is to assign radio frequencies to transmitters such that transmitters that are close (distance 2 apart) to at different locations without causing interference. The each other receive different frequencies and transmitters problem is closely related to graph coloring where the that are very close together (distance 1 apart) receive fre- vertices of a graph represent the transmitters and adja- quencies that are at least two apart. To keep the frequency cencies indicate possible interferences. In1 Griggs and bandwidth small, frequencies that have been assigned to

*Author for correspondence Upper Bound for the Radio Number of Some Families of Sunlet Graph

the radio network. The minimum frequencies assigned is Definition 2.5. radio number (rn). Let G be a graph with vertex set V(G) and edge set Subsequently, different bounds of rn were obtained for E(G). The Middle graph of G, denoted by M(G), is defined various graphs. A common parameter used is ∆ , the maxi- as follows. The vertex set of M(G) is V(G) E(G). Two mum degree of a graph. The obvious lower bound for rn is vertices x,y in the vertex set of M(G) are adjacent in M(G) 1 ∆ +1, achieved for the tree K1,n. In it was shown that in case one of the following holds: for every graph G, rn≤∆+∆2 2 . This upper bound was (i) x,y are in E(G) and x,y are adjacent in G. later improved to rn ≤∆+∆2 in 2. (ii) x is in V(G), y is in E(G), and x,y are incident in G 13. For some classes of graphs, tight bounds are known . and can be computed efficiently. These include paths, Definition 2.6. cycles, wheels and complete k-partite graphs,trees2, Let G be a graph with vertex set V(G) and edge set cographs2, interval graphs2, unicycles and bicycles3, out- E(G). The Total graph of G, denoted by T(G), is defined erplanar planar graphs3, hexagons-meshes4, hexagons as follows. The vertex set of T(G) is V(G) E(G). Two and unit interval graphs5, hypercubes6,7, bipartite graphs8, vertices x,y in the vertex set of T(G) are adjacent in T(G) k-almost tress9, cacti, and grids, and Sunlet graphs10. in case one of the following holds: In this paper, we extend the upper bounds of rn to (i) x,y are in E(G) and x,y are adjacent in G. sunlet families of graphs. Other types of graphs have also (ii) x,y is in V(G), and x,y are adjacent in G. been studied, and their bounds are given in this paper. (iii) x is in V(G), y is in E(G), and x,y are incident in All graphs considered in this paper are finite, nontrivial, G 13. simple, undirected and connected. Definition 2.7. The central graph of a graph G, C(G) is obtained by sub- 2. Preliminaries dividing each edge of G exactly once and joining all the 14 Definition. 2.1 non adjacent vertices of G . A k-vertex coloring of G is an assignment of k-colors Analysis: 1,2,....k to the vertices of G . The coloring is proper if no By the analysis of various graph families to obtain two distinct adjacent vertices have the same color. G is radio number, it is found that on radio coloring of a graph k-vertex colorable if G has a proper k-vertex coloring. The using positive integers, the radio number is obtained by chromatic number χ(G) of G is the minimum k for which using either only odd integers or even integers. The usage G is k-colorable 11. of both odd and even integers alternatively for coloring the graphs gives maximum rn compared to the above Definition 2.2. method. A k-radio coloring of a graph is a function f from the vertex set V(G) to the set of all nonnegative integers 3. Results 1,2,…,k such that (i) f (x)−≥ f (y) 1 if d(x,y)=2 and Result: 1 The radio chromatic number of path P is ∆+1 for (ii) f (x)−≥ f (y) 2 if d(x,y)=1 n n ≥ 310. The radio number of G is the smallest k for which G Result: 2 has radio coloring and is denoted by rn(G)12. The radio chromatic number of comb graph G is ∆+1 for n ≥ 310 . Definition 2.3. Theorem: 3 The n-sunlet graph is a graph on 2n vertices obtained The radio chromatic number of star graph K is by attaching a n-pendant edges to the vertices of cycle C 1,n n ∆+1 10 . and is denoted by S 13 n . Theorem: 4 Definition 2.4. The radio chromatic number of cycle10 The Line graph of a graph G, denoted by L(G), is a ∆+1 n ≡ 0 mod 3 graph whose vertices are the edges of G, and if u, v∈ E(G)  rn( Cn )=∆+ 2 n ≡ 1mod 3 then u, v∈ E(L(G)) if u and v share a vertex in G 13. ∆+3 n ≡ 2 mod 3.

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Theorem: 5 Proof: The radio chromatic number of Sunlet graph S n Let us deﬁne the vertex set V and the edge set E of Sn » ∆+2 n ≡ 1mod3 as V(Sn) = {v1,...,vn} {u1,...,un} where vi are the vertices rn(Sn ) =  of cycles taken in cyclic order and u are the pendent ∆+1 Otherwise i vertices such that viui is a pendent edge and E(Sn) = {e′i

3.1 Structural properties of Sunlet graph : 1≤i≤ n}»{ei : 1 ≤ i ≤ n−1}»{en}, where ei is the edge

vivi+1 (1 ≤ i ≤ n−1), en is the edge vnvl and e′i is the edge • Number of Vertices in Sn is P = 2n viui (1 ≤ i ≤ n). • Number of Edges in Sn is q = 2n By the deﬁnition of middle graph • Maximum degree in Sn is ∆ = 3 V(M(Sn))=V(Sn)»E(Sn)={vi:1≤i≤n}»{ui:1≤i≤n}»{v′i • Minimum degree in Sn is d = 1 :1 ≤ i ≤ n}»{u′i :1 ≤ i ≤ n},where v′i and u′i represents the ′ 3.2 Structural properties of Line graph of edge ei and e i (1≤ i ≤ n) respectively. Sunlet graph Consider the following 9-coloring of (1,3,5,7,9,11,13,15,17) of M(Sn). • Number of Vertices in L(Sn) is P = 2n Assign the color 1 to v1, 3 to v′1, 5 to v2, 7 to v′2, 9 to v3, • Number of Edges in L(Sn) is q = 3n 11 to v′3, 5 to u1, 13 to u′1, 1 to u2, 15 to u′2, 3 to u3, 17 to u′3. • Maximum degree in L(Sn) is ∆ = 4 Note: • Minimum degree in L(Sn) is d = 2 Every vertex in the cycle (ie) v1,...vn, v′1,..., v′n can be 3.3 Structural properties of Middle graph of reached from the remaining vertices in the cycle by dis- Sunlet graph tance two .So, they are colored using different colors. • Number of Vertices in M(S ) is P = 4n ′′ n cvvvv(1 ,...n ,1 ,...n )= 2 n • Number of Edges in M(Sn) is q = 7n

• Maximum degree in M(Sn) is ∆ = 6 Then the remaining 2n vertices are colored in such a

• Minimum degree in M(Sn) is d = 1 way that pendant vertices are colored using repeated colors and vertices with degree 3 is colored with n different 3.4 Structural properties of Total graph of colors. Sunlet graph Lemma: 7 • Number of Vertices in T(Sn) is P = 4n The radio chromatic number of Total graph of Sunlet • Number of Edges in T(S ) is q = 9n n graph Sn is ∆+3 for n=3. • Maximum degree in T(Sn) is ∆ = 6 Proof: • Minimum degree in T(Sn) is d = 2 The vertex set and edge set of nS are as described in 3.5 Structural properties of Central graph lemma 6. of Sunlet graph By the deﬁnition of total graph V(T(Sn))=V(Sn)»E(Sn)={vi :1≤ i≤ n}»{v′i:1≤i≤n}»{ui • Number of Vertices in C(Sn) is P = 4n 2n: 1 ≤ i ≤ n}»{u′i : 1 ≤ i ≤ n},where v′i and u′i represents the 2n+ • Number of Edges in C(Sn) is q =  edge e and e′ (1≤ i ≤ n) respectively. 2 i i • Maximum degree in C(Sn) is ∆ = 6 Consider the following 9-coloring of • Minimum degree in C(S ) is d = 2 n (1,3,5,7,9,11,13,15,17) of T(Sn). Assign the color 1 to v , 3 to v′ ,5 to v ,7 to v′ ,9 to v , 4. Radio Number of Line, Middle, 1 1 2 2 3 11 to v′3, 7 to u1, 13 to u′1, 11 to u2, 15 to u′2, 3 to u3, and 17

Total and Central Graphs of to u′3. Sunlet Graph. The above mentioned coloring is radio coloring. Theorem: 8 Lemma: 6 The radio chromatic number of Middle graph of For n≥3, the radio chromatic number of line graph of sunlet graph L(Sn)is Sunlet graph Sn is ∆+3 for n=3.

Vol 9 (46) | December 2016 | www.indjst.org Indian Journal of Science and Technology 3 Upper Bound for the Radio Number of Some Families of Sunlet Graph

∆+1 n ≡ 2mod 3 5 to v′2, u′5, v′7,… ∆+1 n ≡ 0(mod 6), n ≡ 0(mod 10) 7 to u′ , v′ , u′ ,…  3 5 8 rn( L ( S ) )= ∆+ 2 n ≡ 0mod (25+10i), i=0,1,2,... n  9 to v′3, u′6, v′8,… ∆+2 n ≡ 0mod 2, except n=5i&6i, i=1,2,..  (Note: 1,3,5,7,9 are consecutively assigned to vertices ∆+2 n ≡ 0mod 3,n ≡ 1mod 3 except n=6i, i=1,2,... v′i and u′i (ie) if c1 is assigned to v′1 and 3 to u′2 and so on) Proof: Case(iii): The vertex set and edge set of nS are as described in

lemma 6. rn(L(Sn))=∆+2 n≡0 mod(25+10i), i=0,1,… By the deﬁnition of line graph V(L(S )) = E(S ) = {u′ n n i Consider the following 6-coloring of (1,3,5,7,9,11) of :1 ≤ i ≤ n}»{v′i : 1 ≤ i ≤ n−1}»{v′n} where v′i and u′i repre- L(Sn). sent the edge ei and e′i (1 ≤ i ≤ n) respectively. Sub case(i): n=3k+1, k=8+10i, i=0,1,… Case(i): rn(L(S ))=∆+1 n≡2 mod3 n Assign the color Consider the following 5-coloring of (1, 3, 5, 7, 9) of 1 to v′ , v′ , v′ , …,v′ and u′ L(S ). 1 4 7 n-3 n-1 n 3 to v′ , v′ , v′ ,…,v′ Assign the color 2 5 8 n-2 5 to v′3, v′6, v′9,…,v′n-1 1 to v′1, v′4, v′7, v′10,…,v′n-4 and u′n-1 7 to u′3, u′5, u′7,…,u′n-3 and v′n 3 to v′2, v′5, v′8,…,v′n-3 and u′n 9 to u′2, u′4, u′6,…,u′n-2 5 to v′3, v′6, v′9,…, v′n-2 and u′1 11 to u′1 7 to u′2, u′4, u′6,…,u′n-3 and v′n-1 Sub case(ii): n=3k+2, k=11+10i, i=0,1,… 9 to u′3, u′5, u′7,…, u′n-2 and v′n Sub case(iii): n=3k, k=15+10i, i=0,1,… Case(ii): rn(L(S ))=∆+1 n≡0 mod6, n≡0 mod10 n For Sub cases (ii) and (iii) the colors are assigned in a Consider the following 5-coloring of (1,3,5,7,9) of suitable way as in sub case (i) L(Sn). Case(iv): Sub case(i): rn(L(S ))=∆+1 n≡0 mod6 n rn(L(S ))=∆+2 n≡0 mod2 expect n=5i and6i Assign the color n Consider the following 6-coloring of (1,3,5,7,9,11) of 1 to v′1, v′4, v′7, v′10,…, v′n-2 L(Sn). 3 to v′2, v′5, v′8,…,v′n-1 Sub case(i): n≡2 mod3 5 to v′3, v′6, v′9,…,v′n Assign the color 7 to u′2, u′4, u′6,…, u′n 1 to v′ , v′ , v′ , …,v′ and u′ 9 to u′1, u′3, u′5,…,u′n-1 1 4 7 n-4 n-1 3 to v′ , v′ , v′ ,…, v′ and u′ Sub case(ii): rn(L(S ))=∆+1 n≡0 mod10 2 5 8 n-3 n n 5 to v′ , v′ , v′ ,…, v′ and u′ Assign the color 3 6 9 n-2 1 7 to u′2, u′4, u′6,…, u′n-4 andv′n-1 1 to v′1, u′4, v′6,… 9 to u′3, u′5, u′7,…, u′n-3 and v′n 3 to u′2, v′4, u′7,… 11 to u′n-2 Sub case(ii): n≡1 mod3 Assign the color

1 to v′1, v′4, v′7, …,v′n-3

3 to v′2, v′5, v′8,…, v′n-2

5 to v′3, v′6, v′9,…,v′n-1

7 to u′3, u′5, u′7,…, u′n-3 and v′n

9 to u′2, u′4, u′6,…,u′n

11 to u′1, u′n-1 Case(v):

rn(L(Sn))=∆+2 n≡0 mod3, n≡1 mod3 Consider the following 6-coloring of (1,3,5,7,9,11) of L(S ). Figure 1. Line graph of Sunlet graph L(Sn). n

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Sub case(i): n≡0 mod3 except n=6i (c) The vertices {ui :1 ≤ i ≤ n} are colored by Assign the color 1, 3, and 5 1 to v′ , v′ , v′ , v′ ,…, v′ 1 4 7 10 n-2 Here u is assigned 1, and u as 5. 3 to v′ , v′ , v′ , …, v′ 1 n 2 5 8 n-1 An easy check shows that this coloring is radio 5 to v′ , v′ , v′ ,…,v′ 3 6 9 n coloring. 7 to u′2, u′4, u′6,…,u′n-1 Theorem: 10 9 to u′3, u′5, u′7,…,u′n Let n ≥ 5, then the radio chromatic number of Total 11 to u′1 graph of sunlet graph S is Sub case(ii): n≡1 mod3 when n is prime n The colors are assigned in a suitable way as in subcase (i) ∆+1 n ≡ 0mod 10 ∆+2 n ≡ 0mod 5 rn(() T Sn ) =  Theorem: 9 ∆+2 n ≡ 0mod (k+5i), i=0,1,2,... & k=8,9 Let n ≥ 4, then the radio chromatic number of Middle  ∆+3 n ≡ 0mod (k+5i), i=0,1,2,... & k=6,7 graph of sunlet graph Sn is ∆+2 . Proof: Proof: The vertex set and edge set of S are as described in The vertex set and edge set of S are as described in n n lemma 6. lemma 6. By the deﬁnition of total graph By the deﬁnition of middle graph V(T(Sn))=V(Sn)»E(Sn)={vi :1≤ i ≤ n}»{v′i : 1 ≤ i ≤ V(M(G))=V(Sn)»E(Sn) ={vi:1≤i≤n}»{ui:1≤i≤n}»{v′i n}»{ui : 1 ≤ i ≤ n}»{u′i : 1 ≤ i ≤ n},where v′i and u′i repre- :1 ≤ i ≤ n}»{u′i :1 ≤ i ≤ n}, where v′i and u′i represents the sents the edge ei and e′i (1 ≤ i ≤ n) respectively. edge ei and e′i (1≤ i ≤ n) respectively. Consider the following 8-coloring of (1,3,5,7,9,11,13,15) Case(i) of M(Sn). rn(T(Sn))=7, for n≡0mod10 Assign the color in the following manner Consider the 9-coloring (1,3,5,7,9,11,13,15,17) of T(S ). (a) The vertices {v :1≤i≤n} and {v′ :1 ≤ i ≤ n} are colored n i i Assign the color using the following patterns (a) 1 to v , k=1+5i, i=0,1,2,… 1,3,5,7 k 1 to v′ , k=3+5i, i=0,1,2,… 1,9,3,5 k (b) 3 to v , k=4+5i, i=0,1,2,… 1,7,3,9 k 3 to v′ , k=1+5i, i=0,1,2,… 1,5,3,7 k (c)5 to vk, k=2+5i, i=0,1,2,… Here v′n is assigned 11, v′1 as 3 and v1 as 1 5 to v′k, k=4+5i, i=0,1,2,… (b) The vertices {u′i :1 ≤ i ≤ n} are colored by (d) 7 to vk, k=5+5i, i=0,1,2,… 11, 13, and 15 7 to v′k, k=2+5i, i=0,1,2,…

(e) 9 to vk, k=3+5i, i=0,1,2,…

9 to v′k, k=5+5i, i=0,1,2,…

11 to ui, when i is even

11 to u′i, when i is odd

13 to ui, when i is odd

13 to u′i, when i is even Case(ii)

rn(T(Sn))=8, for n≡0 mod5 The coloring (a) to (e) of above case follows here. Then assign

11 to ui, when i is even

11 to u′i, when i is odd

Figure 2. Middle graph of Sunlet graph M(Sn). 13 to ui, when i is odd

Vol 9 (46) | December 2016 | www.indjst.org Indian Journal of Science and Technology 5 Upper Bound for the Radio Number of Some Families of Sunlet Graph

13 to u′i, when i is even The remaining vertices with degree 2 are assigned

15 to u′n with some colors in an easy manner. Case(iii) Case(iv)

rn(T(Sn))=8, for n≡0 mod(k+5i) i=0,1,2,and K=8,9 rn(T(Sn))=9, for n≡0mod(k+5i) i = 0,1,2, and K = 6,7 The coloring (a) to (e) of above case follows here. The coloring (a) to (e) of above case follows here. Subcase(i) For k=8 Subcase(i) For k=6 when n is even assign when n is even assign

11 to v′n 11 to vn–1, 13 to v′n

13 to u′i, when i is odd 15 to u′1, u′n–1

15 to u′i, when i is even 11 to u′i, when i is even

The remaining vertices v′i with degree 2 are assigned 13 to u′i, when i is odd

with some colors in an easy manner. 11 to u′n when n is odd assign The remaining vertices with degree 2 are assigned

11 to v′n and u′i, i is odd except v1 with some colors in an easy manner.

13 to u′i, when i is odd except u′1 when n is odd assign

15 to v2, vn 11 to vn–1, c13 to v′n

The remaining vertices with degree 2 are assigned 15 to u1, u′n–1

with some colors in an easy manner. 11 to u′i, when i is odd except u′n–3 13 to u′ , when i is even except u′ Subcase(ii) For k=9 i n–2 17 to u′ when n is odd assign n The remaining vertices with degree 2 are assigned 11 to v′ , 13 to v , 15 to v′ n-1 n n with some colors in an easy manner. 9 to u′1, 1 to u′n-1, 5 to u′n

11 to u′i, when i is even Subcase(ii) For k=7

13 to u′i, when i is odd when n is odd assign

The remaining vertices with degree 2 are assigned 9 to u′1, 1 to u′n-1, 1 to u′n

with some colors in an easy manner. 11 to vn-1, 13 to v′n-1

when n is even assign 15 to vn, 17 to v′n

11 to v′n-1, 13 to vn, 15 to v′n 11 to u′i, when i is odd except u′n-4

9 to u′1, 1 to u′n-1, 5 to u′n 13 to u′i, when i is odd except u′n-3

11 to u′i, when i is even 15 to u′n-2

13 to u′i, when i is odd The remaining vertices with degree 2 are assigned

15 to u′n-2 with some colors in an easy manner.

(Note: for n=7, u′n-3=11) when n is even assign

11 to vn-1, 13 to v′n-1

15 to vn, 17 to v′n

9 to u′1, 1 to u′n-1, 5 to u′n

11 to u′i, when i is even except u′n-3

13 to u′i, when i is odd except u′n-2

15 to u′n-2 The remaining vertices v′ with degree 2 are assigned with some colors in an easy manner. Theorem 11. For n≥ 4 the radio chromatic number of Central graph of Sunlet graph 15 ∆+4 n is even rn(C(S )) = Figure 3. Total graph of Sunlet graph T(S ). n  n ∆+5 n is odd

6 Vol 9 (46) | December 2016 | www.indjst.org Indian Journal of Science and Technology A.Vimala Rani and N. Parvathi

Proof. 3. Jonas K. Graph Colorings analogues with a condition at distance two; L(2,1)-labelings and list λ -labelings, Ph.D. The vertex set and edge set of nS are as described in lemma 6. thesis, University of South Carolina.1993. By the definition of Central graph 4. Bertossi AA, Pinotti C, Tan R.B. L(2,1,1)-Labeling problem V(C(S )) =V(S )∪v′ ∪u′ = {v : 1 ≤ i ≤ n}∪{v′ : 1 ≤ i ≤ on graphs, In preparation (1999). n n i i i i 5. Sakai D. Labeling chordal graphs: distance two condition, n∪{u :1 ≤ i ≤ n}∪{u′ : 1 ≤ i ≤ n} where v′ represent the i i i SIAM J.Disc.Math. 7(1194) .p. 133–140. vertices subdiving v ,v (1 ≤ i ≤ n–1), and u′ represent the i i+1 i, 6. Georges JP, Mauro DW. On the size of graphs labeled with a vertices subdiving u ,v (1 ≤ i ≤ n) respectively. i i condition at distance two. Journal of Graph Theory 22 1996. Case 1: rn(C(Sn)) = ∆ + 4 for n is even p. 47–57. The following ∆+4 coloring for C(Sn) admits radio 7. Georges JP, Mauro DW, Whittlesey MA. Relating path cov- erings to vertex labeling with a condition at distance two, coloring. For (1 ≤ i ≤ n) assign the color i to vi, and for(1 ≤ i ≤ n) assign the color n+i to u . Discrete Mathematics 135. 1994. p.103–11. i 8. Yeh RK, Labelling graphs with a condition at distance two. Assign the color 2n+1 to u′ (1 ≤ i ≤ n), for (1 ≤ i ≤ n–1) i Ph.D Thesis, University of South Cardino(1990). 2n+2 to v′ if i isodd, and 2n + 3 to v′ if i is even. i i 9. Fiala J, Kloks T, Kratochvil J. Fixed-parameter com- Case 2:rn(C(Sn)) = ∆ + 5 for n is odd plexity of λ -labelings, In: Graph-Theoretic Concept of The following ∆+5 coloring for C(Sn) admits radio Computer Science, Proceedings 25th WG’99, Lecture Notes in Computer Science Vol.1665, Springer Verlag (1999). p. coloring. For (1 ≤ i ≤ n) assign the color i to vi, and for (1 ≤ i ≤ n) assign the color n+i to u . 350–63. i 10. Vimala Rani A , Parvathi N. Radio number of Star Assign the color 2n+1 to u′ (1 ≤ i ≤ n), for (1 ≤ i ≤ i graph and Sunlet graph, Global Journal of Pure and n–1) 2n+2 to v′ if i is odd , 2n + 3 to v′ if i is even, and i i Applied Mathematics (GJPAM) ISSN 0973-1768 . 2016; 2n + 4 to v′ . n 12( 1):151–56. 11. Bondy JA , Murty USR. Graph Theory with Applications. 5. Conclusion 12. Chang J, David Kuo, SIAM J. Discrete Math., The L(2,1)- Labeling Problem on Graphs, Gerard 9(2), 309–316. In this paper we obtain the radio chromatic number for 13. Vernold Vivin J, Vekatachalam M. On b-chromatic number Line, Middle, Total and Central graphs of Sunlet graph and of Sun let graph and wheel graph families, Journal of the improved bounds of radio chromatic number of Sunlet Egyptian Mathematical Society 2015; 23, p. 215–18. graph. Despite of the few results in radio coloring we still 14. Thilagavathi K, Shanas Babu P. A Note on Acyclic Coloring experiment on radio number of various families of graphs of Central Graphs. International Journal of Computer Applications. 0975-8887. 2010; September, 7(2). and thuis radio coloring is useful in channel assignment 15. Ramachandran M, Parvathi N. The Medium Domination problem which has wide range of applications16. Number of Jahangir Graph Jm,n, Indian Journal of Science and Technology. 2015; March, 8(5): 400–6. 6. References 16. Pounambal M. Survey on Channel Allocation Techniques for Wireless Mesh Network to Reduce Contention with 1. Griggs, Yeh RK. Labeling graphs with a condition at dis- Energy Requirement, Indian Journal of Science and tance 2, SIAM Journal of Discrete Mathematics. 5.1992. Technology. 2016; August, 9(32). p.586–95. 17. Kalfakakou R, Nikolakopoulou G, Savvidou E, Tsouros 2. Chang GJ, Kuo D. The L(2,1)-Labeling problem on graphs, M. Graph Radiocoloring Concepts , Aristotle University of SIAM J. Disc .Math. 9 1996. p.309–16. Thessaloniki, Faculty of Engineering Thessaloniki, Greece.

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