Triple Domination Number and it s Chromatic Number of Graphs

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1 Triple Domination Number and it s Chromatic Number of Graphs A.Nellai Murugan 1 Assoc. Prof. of Mathematics, V.O.Chidambaram, College, Thoothukudi , Tamilnadu, India anellai.vocc@gmail.com G.Victor Emmanuel 2 Research Scholar, SS Pillai Research Centre for Mathematics,VOC College, Thoothukudi , Tamilnadu, India vvj_s18@yahoo.com Abstract In a graph G, a vertex dominates itself and its neighbours. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number γ(g) is the minimum cardinality of a dominating set. A set S is called a Triple dominating set of a graph G if every vertex in V is dominated by at least three vertices in S. The minimum cardinality of a Triple dominating set is called Triple domination number of G and is denoted by Tγ(G). The connectivity (G) of a connected graph G is the minimum number of edges whose removal results in a disconnected or trivial graph and the chromatic number χ(g) is defined as the minimum n for which G has a n-coloring. In this paper we find an upper bound for the sum of the Triple domination number and chromatic number of a graph and characterize the corresponding extremal graphs. Mathematics Subject Classification: 05C69 Key Words: Domination, Domination number, Triple domination, Triple domination number, edge connectivity and the chromatic number. I.INTRODUCTION: The concept of domination in graphs evolved from a chess board problem known as the Queen problem- to find the minimum number of queens needed on an 8x8 chess board such that each square is either occupied or attacked by a queen. C.Berge[12] in 1958 and 1962 and O.Ore[11] in 1962 started the formal study on the theory of dominating sets. Thereafter several studies have been dedicated in obtaining variations of the concept. The authors in [2] listed over 1200 papers related to domination in graphs in over 75 variation. The graph G = (V, E) we mean a finite, undirected and connected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m respectively. The degree of any vertex u in G is the number of edges incident with u is denoted by d(u). The minimum and maximum degree of a graph G is denoted by δ(g) and ( ) respectively. For graph theoretic terminology we refer to Chartrand and Lesniak[1] and Haynes et.al[2,3]. Let v ЄV, The open neighbourhood and closed neighbourhood of v are denoted by N(v) and N[v] = N(v) {v} respectively. If S then N(S) = ( ) for all vєs and N[S] = N(S) S. If S and uєs then the private neighbour set of u with respect to S is defined by pn[u, S] = {v : N[v] S ={u} }. We denote a cycle on n vertices by C n, a path on n vertices by P n and a complete graph on n vertices by K n. A bipartite graph is a graph whose vertex set can be divided into two disjoint sets V 1 and V 2 such that every edge has one end in V 1 and another in V 2. A complete bipartite graph is a bipartite graph where every vertex of V 1 is adjacent to every vertex in V 2. The complete bipartite graph with partitions of order =, and = is denoted by K m,n. A wheel graph, denoted by W n is a graph with n vertices formed by connecting a single vertex to all vertices of C n-1. H{m 1, m 2,...,m n ) denotes the graph obtained from the graph H by pasting m i edges to the vertex v i ЄV(H), 1 i n, H(P m1, P m2..,p mn ) is the graph obtained from the graph H by attaching 1

2 the end vertex of P mi to the vertex v i in H, 1 i n. Bistar B(r,s) is a graph obtained from K 1,r and K 1,s by joining its centre vertices by an edge. In a graph G, a vertex dominates itself and its neighbours. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number γ(g) is the minimum cardinality of a dominating set. Harary and Haynes[4] introduced the concept of double domination in graphs. A set S is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of double dominating set is called double domination number of G and is denoted by dd(g). A edge cut, or separating set of a connected graph G is a set of edges whose removal results in a disconnected graph. The connectivity or edge connectivity of a graph G denoted by (G) of a connected graph G is the minimum number of edges whose removal results in a disconnected or trivial graph. A coloring of a graph is an assignment of colors to its points so that no two adjacent points have the same color. The set of all points with any one color is independent and is called a color class. An n-coloring of a graph G uses n-colors it thereby partitions V into n color classes. The chromatic number χ(g) is defined as the minimum n for which G has an n-coloring. A graph G is n- colorable if ( ). A graph G is n-chromatic if ( ) =. Since G obviously has a p-coloring and ( )- coloring it must also have an n-coloring whenever ( ) < <. A graph is 1-chromatic iff it is totally disconnected. The chromatic numbers of the familiar graphs are χ( )=n, χ( )=n-1, χ( )=1, χ(, )= 2, χ( )=2, =3, χ( )=2. A matching M is a subset of edges so that every vertex has degree at most one in M. Several authors have studied the problem of obtaining an upper bound for the sum of a domination parameter and a graph theoretic parameter and characterized the corresponding extremal graphs. J.Paulraj Joseph and S. Arumugam[5] proved that γ(g) + κ(g) n and characterized the corresponding extremal graphs. In this paper, We obtained an upper bound for the sum of the Triple domination number and chromatic number of a graph characterized the corresponding extremal graphs. We use the following theorems. Theorem 1.0 [2] For any graph G, dd(g) n Theorem 1.1 [15] For any graph G, Tγ(G) n Theorem 1.2 [1] For any graph G, ( ) ( ) ( ). Theorem 1.3 [14 ] A graph is bi colorable iff it has no odd cycle. Theorem 1.4 [14] For any graph G, ( ) 1 + ( ) where the maximum is taken over all induced subgraphs G of G. Corollary 1.5 [14]For any graph G, The chromatic number is at most one greater than the maximum degree that is ( ) 1+. The connectivity of a disconnected graph is 0 while the connectivity of a connected graph with a cut point is 1.The line connectivity = ( ) of a graph G is the minimum number of lines whose removal results in a disconnected or trivial graph, Thus ( ) =0. The line connectivity of a disconnected graph with a bridge is 1 Theorem 1.6 [14] If G is of degree r, then ( ) = r in particular ( ) = 1 Corallary 1.7 [14] The maximum line connectivity of a (n,m) graph equals the maximum connectivity. II.MAIN RESULT Definition 2.0 [15] A set S V(G) is a Triple dominating set for G if every vertex in V is dominated by at least three vertices in S. The minimum cardinality of Triple dominating set is called Triple domination number of G and is denoted by T ( ). The line connectivity = ( ) of a graph G is the 2

3 minimum number of lines whose removal results in a disconnected or trivial graph, Thus ( ) =0. The line connectivity of disconnected graph with a bridge is 1. The chromatic number χ(g) is defined as the minimum n for which G has an n-coloring. A graph G is n-chromatic if χ(g) = n. Lemma 2.1 For the graph K n, n 3, T ( ) 3 and ( ) = Lemma 2.2 For the graph K n, 3,We have T ( )+ ( ) 2. Example 2.3 For the graph K 3, T ( ) =3 and ( ) =3,We have T ( )+ ( ) = 6. Lemma 2.4 For the graph K n, We have T ( )+ ( ) = 2 for n=3. Lemma 2.5 For the graph K n, We have T ( )+ ( ) =2 for i= n-3 and n 4. Example 2.6 For the graph K 4, T ( ) =3 and ( ) =4,We have T ( )+ ( ) = 7. Example 2.7 For the graph K 5, T ( ) =3 and ( ) =5,We have T ( )+ ( ) = 8. Example 2.8 For the graph K 6, T ( ) =3 and ( ) =6,We have T ( )+ ( ) = 9. Example 2.9 For the graph K 7, T ( ) =3 and ( ) =7,We have T ( )+ ( ) = 10. Lemma 2.10 For the graph K n,we have T ( )+ ( ) 2 1 for n 4. Theorem 2.11 For any connected graph G, T ( )+ ( ) = 2 iff G is isomorphic to K 3 or K 4 -M with =1 Proof: T ( ) + ( ) +1+ = +1+ = +1+ 1=2 if = and we have T ( )=n, ( ) = then G is a complete graph on n vertices, Since T ( )=3 we have n=3 and ( ) = 3. Hence G is isomorphic to K 3 or K 4 -M where M is a matching with =1 given fig:1a and 1b. fig:1a fig:1b T ( ) =3, ( ) =3 T ( ) =3, ( ) =3 Theorem 2.12 For any connected graph G, T ( )+ ( ) =2 1 iff G is isomorphic to K 4, K 4 -M, or C 4 where M is a matching with = 1 2 Proof: Let T ( )+ ( ) =2 1 then there are two cases to be considered (i) T ( ) = 1 and ( ) = (ii) T ( ) = and ( ) = 1 Case 1: (i) T ( )= 1 and ( ) =, Then G is a complete graph on n vertices, Since χ( )=n, we have n=4 and T ( ) =3. Hence G is isomorphic to K 4. Case 2: (ii) T ( ) = and ( ) = 1, Since we have < and we know that 1, Hence we have =. If and we know that ( ) 1+ and ( ) 1 which implies ( ) 1, 1 1, 2. Suppose we have = 2, = 2. Then T ( )= 3 4, Hence n= 3or 4. If n=3 we have T ( )=3 and ( ) =2 then G is isomorphic to K 4 -M where M is a matching with = 2.Which is a contradiction. If n=4 we have T ( ) 4 and ( ) =3. Then G is isomorphic to K 4 -M where M is a matching with = 1 with T ( )=3. 3

4 Theorem 2.13 For any connected graph G, T ( )+ ( ) =2 2 iff G is isomorphic to K 5, K 5 -M, K 4 -M, K 4, K 4 + K 2, C 4 where M is a matching with =1. Proof: Let T ( )+ =2 2 then there are three cases to be considered(i) T ( ) = 2 and ( ) = (ii) T ( ) = 1 and ( ) = 1 (iii) T ( ) = and ( ) = 2 Case 1: T ( ) = 2 and ( ) =, Then G is a complete graph on n vertices, Since χ( )=n, We have n=5,t ( ) =3 and ( ) =5. Hence G is isomorphic to K 5. Case 2: T ( )= 1 and ( ) = 1, we have = 2, 2, Then there are two vertices are not adjacent then we can have two vertices are of identical color. Hence there must be n-1 colors. Hence ( ) = 1and n=4 or 5, we have T ( ) =3 or 4. SubCase 2.1: Suppose n=4, We must have T ( ) =3 and ( ) =3, then G is isomorphic to K 4 -M where M is a matching with =1by fig:1c fig:1c fig:1d =2, T ( ) =3, ( ) =3 =3, T ( ) =4, ( ) =4 SubCase 2.2: Suppose n=5, We must have T ( ) =4 and ( ) =4, then G is isomorphic to K 5 -M where M is a matching with =1by fig:1d. Case 3 : T ( ) = and ( ) = 2, we have = 3and there fore 3, Then G is isomorphic to K n -M where M is a matching in K n and T ( ) 5 or 6. There fore we must have n=5 or 6 and ( ) 3 or 4. Suppose n=5 we have T ( ) 5 and ( ) =3 with = 3, since of a disconnected graph results when all the lines incident with a point of minimum degree are removed. Let be the edge cut of G with = = 3, and let ={,,,, },,,. = and ( ) = +1={C 1,C 2, C n-3+1 } or { C 1,C 2, } Subcase 3.1: =, Then there are two cut edges incident with and atmost one vertex in G, Since there are two vertices which are not adjacent in K 4 -e has at most one color is repeated,hence we can have 3 vertices are of different color and one vertex is of repeated color and is having the color which is non adjacent in. Hence we have ( ) =3, There are colors which repeats number of times, then there exists v 1 C 1, v 2 C 2, v 3 C 3, such that {v 1,v 2,v 3 } is a Triple dominating set of G with ( ) =3by fig:2. fig:2 fig:2a =2, T ( ) =3, ( ) =3 =2, T ( ) =4, ( ) =4 Subcase 3.2: =, There are two cut edges incident with and at most one vertex in G, Since = is a complete graph on four vertices then must have four colors by definition of coloring and G must have four colors in which at most one color is repeated. Hence 4

5 ( ) =4. If v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set T ( ) = 4 which is a contradiction to ( ) =3 by fig :2a. Subcase 3.3: = or K 2,3, There are two cut edges incident with and atmost one vertex in G. Since =C 4 it must have two colors and the cut edges must incident on two different colors. Hence ( ) =3 and T ( ) = 3 by fig:3a otherwise G is isomorphic to C 4 or K 2,3 with T ( ) = 2 and ( ) =2 n-2 by fig:3. which is a contradiction. Suppose n=6, we must have T ( ) 6, Hence T ( ) =4and ( ) =4 with = 3. fig:3 fig:3a fig:3b =2, T ( ) =2, ( ) =2 =2, T ( ) =3, ( ) =3 =3, T ( ) =4, ( ) =4 Subcase 3.4: =, since = with = 3. There exists cut edges belong to such that which is adjacent to atmost three vertices in and one vertex in G, Since has atmost two vertices are non adjacent. There are four vertices of different color and one vertex is of repeated color and is adjacent in G having the color which is non adjacent in. Hence we have ( ) =4. If v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set with T ( )=4 and ( ) =4 by fig:3b Subcase 3.5: = =K 1 +C 4, Then T ( ) 5 and ( ) =4, since = with = 3. There exists cut edges belong to is adjacent to atmost three vertices in since has atmost one vertex is non adjacent and we can have one vertex is of repeated color and since is adjacent to atmost three vertices in and one vertex in G whose color must be different from atleast two adjacent colors. Hence we have ( ) =3 or 4 which is a contradiction to ( ) =3 by fig:4b. Hence we must have ( ) =4 and if v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set with T ( )=4 by fig:4a. If ( )=5 and = then we must have atleast five colors hence ( ) =5 which is impossible to ( ) =2 n-2. fig:4a fig:4b =3, T ( ) =4, ( ) =4 =3, T ( ) =3, ( ) =3 Subcase 3.6: =, Since = 3there are two cut edges incident with and at most one vertex in G which is a contradiction to T ( ) and suppose P 4 K 1 which gives ( ) =3 but it is a contradiction to T ( )=5. Theorem 2.14 For any connected graph G, T ( )+ ( ) =2 3 iff G is isomorphic to K 6, K 5 -e, K 4 -e, K 4. Proof:Let T ( )+ ( ) =2 3, Then there are four cases to be considered(i) T ( ) = 3 and ( ) = (ii) ( ) = 2 and ( ) = 1 (iii) ( ) = 1 and ( ) = 2 (iv) T ( ) = and ( ) = 3 5

6 Case 1: T ( ) = 3 and ( ) =, Then G is a complete graph on n vertices, Since χ( )=n, and since T ( ) = 3 we must have n=6. Hence G is isomorphic to K 6 With T ( )=3, ( ) =6 Case 2: T ( )= 2 and ( ) = 1 and we have = 2 and 2, Then G is isomorphic to K n -M where M is a matching in K n then T ( ) =3 or 4 and if T ( ) =3, ( ) =4 and n=5 then G is either K 5 -e or K 4. But T ( ) =2 3. Hence G is isomorphic to K 5 -e where e is a matching in K 5. If T ( ) =4 then n=6 hence ( ) =4 or 5 and T ( ) = 4 5 then G is isomorphic to K 6 -M where M is a matching in K 6 with =2 or 1 respectively with = 2 that is = 2 given fig:5a,5b,5c. fig:5a fig:5b fig:5c =3, T ( ) =3, ( ) =4 =4, T ( ) =5, ( ) =5 =4, T ( ) =4, ( ) =4 Case 3: T ( )= 1 and ( ) = 2 and we have = 3 and 3. Suppose if = 1 and = n-1 then G is a complete graph on n vertices hence ( ) = which is a contradiction to ( ) = 2. If = 2and = n-2 then G is isomorphic to K n -M where M is a matching on K 4 with =1 or 2 then T ( ) = 3 or 4. If n=4, T ( ) = 3 and ( ) =2 with = 2 which is a contradiction and T ( ) = 3 and ( ) =3 with = 1 which is a contradiction. If n=5, T ( ) =4 and ( ) =4 with = 1 by fig:6a and T ( )=3 and ( ) =3 with = 2 by fig:6b which is a contradiction. Hence = 3 and 3, we have n=5 or 6 since of a disconnected graph results when all the line incident with a point of minimum degree are removed. Let be the edge cut of G with = =n-3 and let ={,,,,, },,,. = and ( ) = +1={C 1,C 2, C n-3+1 } or { C 1,C 2, } fig:6a fig:6b =3, T ( ) =4, ( ) =4 =3, T ( ) =3, ( ) =3 Subcase 3.1 : =,We have T ( ) 4 and ( ) =3 with =n-3 here n=5. Hence there are at most two cut edges incident with two vertices in and one in G. Hence if there exists v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set with T ( )=4 and ( ) =4. Which is a contradiction given fig:2a. Subcase 3.2 : =,we have T ( ) 4 and ( ) =3 with =n-3 here n=5. There are at most two cut edges incident with two in and one in G. Hence if there exists v i = C i, = 1,2,3, and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, such that {v 1,v 2,v 3 } is a Triple dominating set with T ( )=3 and ( ) =3 given fig:2. Subcase 3.3 : = Since =n-3 we have n=5 with ( ) =3 and if there exists v i = C i, = 1,2,3, and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, such that {v 1,v 2,v 3 } is a Triple dominating set with T ( )=3. 6

7 Subcase 3.4 : =, Since =n-3 we must have cut edges and ( ) =5 and T ( ) 5, Hence n=6. which is a contradiction to ( ) =5 2. Subcase 3.5: =, Then we have T ( ) 5 and ( ) =4. Since = with = 3, Then there exists cut edges belongs to which is adjacent to atmost three vertices in and one vertex in G. Hence n=6, Since K 5 -e has at most two vertices are non adjacent. we must have four vertices of different color and one vertex is of repeated color and a vertex is adjacent to having the color which is non adjacent in. Hence we have ( ) =4. if there exists v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set with T ( )=4 by fig:3b. Subcase 3.6: = = +, Then we have T ( ) 5 and ( ) =4. Since = with = 3,There exists cut edges belong to which is adjacent to atmost three vertices in since has atmost one vertex is non adjacent we can have three vertices are at different colors and one vertex is of repeated color and since is adjacent to at most three vertices in and one vertex in G whose color must be different from atleast two adjacent colors. Hence we have ( ) = 3 or 4 which is a contradiction to ( ) = 3. Hence we have ( ) = 4 and if there exists v i = C i, = 1,2,3,4 and there exists vertices v 1 C 1, v 2 C 2, v 3 C 3, v 4 C 4 such that {v 1,v 2,v 3,v 4 } is a Triple dominating set with T ( )=4 by fig:4a. Case 4: (iv) T ( )= and ( ) = 3, we have = 4 and 4, Suppose, If = 1, then = 1, Then G is a complete graph on n vertices, Hence ( ) =, which is a contradiction to ( ) = 3. Suppose if = 2, then = 2, Then G is isomorphic to K n -M where M is a matching in K n. If n=5, T ( ) =5 and ( ) =2 which is impossible with = 4. Hence = n-3 or n-4 we have n = 5 or 6 since of a disconnected graph results when all the line incident with a point of minimum degree are removed. Let be the edge cut of G with = =n-3 and let ={,,, },,,. = and ( ) = +1 ={C 1,C 2, C n-3+1 } or { C 1,C 2, } or Let be the edge cut of G with = 4and let ={,,,,, },,,. = and ( ) = +1={C 1,C 2, C n-4+1 } or { C 1,C 2, }. If n=5 we have = 3. Hence =C 4 then we have ( ) =2 and T ( ) =2 3 which is a contradiction and =K 4 Or K 4 -e, then we have ( ) =3 or 4 n-3 and T ( ) = 3 or 4 which is a contradiction. If n= 6, We have = 4, Suppose if =[K 4 e][1,0,0,0] we have ( ) =3 and T ( ) =3 given fig:7a, But if =K 4 [1,0,0,0], we have ( ) =4 n-3. If = K 5 e we have ( ) =4 n-3 and if = K 5 we have ( ) =5 n-3 both gets contradiction. fig:7a =2, T ( ) =3, ( ) =3 III.CONCLUSION: In this paper we found an upper bound for the sum of Triple domination number and chromatic number of graphs and characterized the corresponding extremal graphs. Similarly Triple domination number with other graph theoretical parameters and graph coloring parameters can be considered. 7

8 REFERENCES: [1] G. Chartrand and Lesniak, Graphs and Digraphs, CRC, (2005). [2] T. W. Haynes, S.T.Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekkar, Inc., New York, (1997). [3] T.W. Haynes, S. T. Hedetniemi and P.J. Slater, Domination in Graphs- Advanced Topics, Marcel Dekker, Inc., New York, (1997). [4] F.Harary and T. W. Haynes, Double Domination in graphs, Ars combin. 55, (2000). [5] J. Paulraj Joseph and S.Arumugam, Domination and connectivity in Graphs, International Journal of management and systems, 8, (1992). [6] C.Sivagnanam, Double Domination Number and Connectivity of Graphs, International Journal of Digital Information and Wireless Communications (IJDIWC) 2(1):40-45, The Society of Digital Information and Wireless Communications, 2012(ISSN X). [7] J. A.Bondy and U. S. R. Murty, Graph Theory, Springer, [8] E. Sampathkumar and, H. B. Walikar, The connected domination number of a graph, J.Math. Phy. Sci., 13(6)(1979), [9] B.D.Acharya, H.B.Walikar, and E.Sampathkumar, Recent developments in the theory of domination in graphs. In MRI Lecture Notes in Math. Mehta Research Instit, Allahabad No.1, (1979). [10]Teresa L. Tacbobo and Ferdinand P.Jamil, Closed Domination in Graphs, International Mathematical Forum, Vol. 7, 2012, No. 51, [11] O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ., Vol.38, Providence, [12] C. Berge, Theory of Graphs and its Applications, Methuen, London, [13] T. Haynes, S. Hedetniemi and M. Henning, Domination in graphs applied to electrical power networks, J. Discrete Math. 15(4), , [14] F.Harary, Graph Theory, Addison-Wisley Publishing Company, Inc. [15] A.Nellai Murugan, G.Victor Emmanuel, Triple Domination Number and Its Connectivity of Complete Graphs, Int. Journal of Engineering Research and Applications, ISSN : , Vol. 4, Issue 1( Version 9), January 2014, pp Author s Biography: Dr. A. Nellai Murugan, Associate Professor, S.S.Pillai Centre for Research in Mathematics, Department of Mathematics, V.O.Chidambaram College, Thoothukudi. College is affiliated to Manonmanium sundaranar University, Tirunelveli-12, TamilNadu. He has thirty years of Post Graduate teaching experience in which ten years of Research experience. He is guiding Five PhD Scholars. He has published more than thirty research papers in reputed national and international journals. Mr.G.Victor Emmanuel, Research Scholar, S.S.Pillai Centre for Research in Mathematics, Department of Mathematics, V.O.Chidambaram College, Thoothukudi. College is affiliated to Manonmanium sundaranar University, Tirunelveli- 12,TamilNadu. He has Thirteen years of Post Graduate teaching experience in which ten years in St.Joseph s College(Autonomous),Trichirappalli-2 and three years in St.Mother Theresa Engineerng College,Tuticorin. He has Published Three research papers in reputed national and international journals. 8