Dominator Coloring of Certain Graphs

International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-2S, December 2018

Dominator Coloring Of Certain Graphs

T.Manjula, R.Rajeswari , Anumita Dey , Krishna Deepika

Abstract: A proper vertex or node coloring of a graph where Snarks [10] play a central role for several well known every vertex of the graph dominates all vertices of some color conjectures in graph theory. class is called the dominator coloring of the graph. The least The Flower snark [10] was introduced by Rufus Isaacs in number of colors used in the dominator coloring of a graph is 1975 and they form an infinite family of snarks. The Double called the dominator coloring number denoted by χ (G). The d Star Snark [10] discovered by Rufus Isaacs is a 30 vertex dominator chromatic number and domination number of closed sun graph, closed helm graph, generalized Flower snark, Double graph with 45 edges. The Watkins Snark [10] discovered by star snark and Watkins snark graph are derived and the relation John J.Watkin in 1989 is a 50 vertex graph with 75 edges. between them are expressed in this paper. They are connected, bridgeless cubic graphs with chromatic index equal to 4 and are non-planar and non-Hamiltonian. Keywords: Coloring, Domination, Dominator Coloring The flower snark Jn [10] has 4n nodes and 6n edges. It is constructed as follows: The n copy of the star graph on 4 I. INTRODUCTION nodes is taken. The central node of each star is denoted by a and the outer nodes by b , c and d . This results in a A dominating set is a subset DS of the vertex or i i i i disconnected graph on 4n nodes with 3n edges (a b , a c and node set of graph G which is such that each node in the i i i i a d for 1 ≤ i ≤ n). graph either belongs to DS or has a neighbour in DS[1]. The i i domination number γ(G) is the cardinality of a smallest  A cycle of length n is constructed by connecting dominating set of G[1]. A proper coloring of a graph G is a the n nodes b1…bn.  Finally a cycle of length 2n is constructed by function : such that for whenever u and v adjacent nodes in G. A dominator connecting the 2n nodes c1, c2...,cn,d1, d2... dn. coloring of a graph G is a proper coloring of graph such that For a flower snark Jn to have the required every node or vertex of G dominates all nodes of at least one properties, n should be odd. The flower snark J3 is also color class. The minimum cardinality of colors used in the known as the Tietze’s graph named after Heinrich Franz graph for dominator coloring is called the dominator Tietze. In this paper the domination number and dominator coloring number denoted by χd (G). [2]. The concept of dominator coloring was introduced by chromatic number of closed Sun graph, closed Helm graph, Ralucca Michelle Gera in 2006 [2]. The relation between Flower snark, Double star snark and Watkins snark is dominator coloring, proper coloring and domination number obtained and a relation between the dominator chromatic of different classes of graphs were shown in [3], [5]. The number, chromatic number and domination number is dominator coloring of prism graph, quadrilateral snake, expressed. triangle snake and barbell graph and M-Splitting graph and M-Shadow graph of Path graph were also studied in various II. DOMINATOR CHROMATIC NUMBER OF CLOSED SUN papers [6], [7], [8]. The algorithmic aspects of dominator GRAPH coloring in graphs have been discussed by Arumugam S A. Proposition : et.al. in [4]. The chromatic number of a closed sun The closed sun graph [9] denoted by with nodes graph : , is given by is constructed as follows: A complete graph with the nodes { } , is surrounded by a cycle with B. Theorem: nodes { }. Then the edges and : (with : ) are added. Every closed sun graph where : , , has A closed Helm graph or a belt graph [11] is constructed dominator chromatic number from the Helm graph by joining its outer vertices. It has 2n+1 vertices 4n edges. It is denoted by . Proof: The closed sun graph with nodes is constructed Revised Version Manuscript Received on 22 December, 2018. T.Manjula, Research Scholar, Professor, Sathyabama Institute of as follows: A complete graph with the nodes { Science & Technology – Deemed to be University, Chennai, Tamil } , is surrounded by a cycle with nodes { nadu,India, (E-mail: [email protected]) }. Then the edges and : (with : R.Rajeswari, Department of Mathematics, Sathyabama Institute of ) are added. Science & Technology – Deemed to be University, Chennai,Tamil nadu, India, ([email protected]). Anumita Dey, Student, Department of ECE, Sathyabama Institute of Science & Technology – Deemed to be University, Chennai, Tamil nadu, India. Krishna Deepika, Student, Department of ECE, Sathyabama Institute of Science & Technology – Deemed to be University, Chennai, Tamil nadu, India.

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Dominator Coloring Of Certain Graphs

Let the node set and edge set of the closed Sun graph D.Corollary: be Every closed Sun graph where :, { } satisfies the relation { } ⌊ ⌋. { } : Proof: By applying theorem 2.2 and lemma 2.3 we get { }. the result. The procedure below explains the dominator coloring of nodes. III. DOMINATOR CHROMATIC NUMBER OF CLOSED HELM For , the nodes are painted with color i. The GRAPH

node is allotted color n. And for the nodes are painted with color respectively. A. Proposition: Then for , the nodes are dominated by color The chromatic number of a closed Helm graph denoted class i respectively. And for , the nodes are by : , is given by dominated by color class respectively. The nodes { . are dominated by color class respectively. Every neighbouring node is given different color and also B. Theorem: it is observed that every node of the graph dominates all the Every closed Helm graph denoted by where nodes of atleast one color class. Thus it is a dominator and n is even, has dominator chromatic number coloring of nodes and the dominator coloring number of ⌈ ⌉ . closed Sun graph where : , is given Proof: by . A closed Helm graph or a belt graph [11] is constructed from the Helm graph by joining its outer vertices. It has 2n+1 vertices and 4n edges. It is denoted by .

Let the node set and edge set of the closed Sun graph be 1 4 푢 { } { } 푢5 5

푣 { } { }

{ }. The procedure below explains the dominator coloring of nodes. 5 푣 푣 2 5 Case 1: When is even and For , the nodes are allotted color 2 when i is odd and color 3 when i is even. The node w is painted with 1

color 1. For ⌈ ⌉ the nodes are allotted ; 푢 푢 3 4 color 3 when i is odd and color 2 when i is even and the 푣 푣4 nodes ; are painted with color respectively. 4 3 The node w and for , the nodes are

dominated by color class 1. Then for ⌈ ⌉ , the

푢 nodes ; ; are dominated by color class 2 respectively.

Every neighboring node is given different color and also Figure 1: Dominator chromatic number of closed Sun it is observed that every node of the graph dominates all the graph is 5. i.e, nodes of atleast one color class. Thus it is a dominator coloring of nodes and the dominator coloring number of C. Lemma: closed Helm graph where is The domination number of closed Sun graph given by where , is given by ⌈ ⌉ : ⌈ ⌉ .

Proof: Case 2: When is even and The node set of the closed Sun graph is For , the nodes are allotted color 2 when i is { }. odd and color 3 when i is even. The node w is painted with Let the dominating set of the closed Sun graph be color 1. For ⌊ ⌋ the nodes ; are pinted with { ⌈ ⌉}. Clearly every node in V-DS is adjacent ; color 3 when i is odd and color 2 when i is even and the to atleast one node of DS and the dominating set DS has the minimum cardinality. Hence the domination number of the closed Sun graph is given by ⌈ ⌉.

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nodes ; are allotted color respectively. The node For , the nodes are allotted color 2 when i is odd and color 3 when i is even. The node w is painted with is painted with color ⌈ ⌉

color 1. For ⌊ ⌋ the nodes are allotted The node w and for , the nodes are ; dominated by color class 1. Then for ⌊ ⌋ , the color 3 when i is odd and color 2 when i is even and the

nodes ; are painted with color respectively. The nodes ; ; are dominated by color class nodes ; and are allotted color 3, the nodes ; , respectively. The node is dominated by color class are painted with color 4 and color ⌈ ⌉ ⌈ ⌉ ; respectively. Every neighboring node is given different color and also it is observed that every node of the graph dominates all the The node w and for , the nodes are nodes of atleast one color class. Thus it is a dominator dominated by color class 1. Then for ⌊ ⌋ , the coloring of nodes and the dominator coloring number of nodes ; ; are dominated by color class closed Helm graph where is respectively. The node is dominated by color class 4 given by Every neighboring node is given different color and also it is

⌈ ⌉ . observed that every node of the graph dominates all the

Case 3: When is even and nodes of atleast one color class. Thus it is a dominator coloring of nodes and the dominator coloring number of For , the nodes are allotted color 2 when i is odd and color 3 when i is even. The node w is painted with closed Helm graph where is given by color 1. For ⌊ ⌋ the nodes ; are allotted ⌈ ⌉ . color 3 when i is odd and color 2 when i is even and the Case 2: When is odd and nodes ; are painted with color respectively. The For , the nodes are allotted color 2 when nodes ; and are allotted color 3 and ⌈ ⌉ i is odd and color 3 when i is even. The nodes and w is respectively. painted with color 4 and 1 respectively. For ⌈ ⌉ The node w and for , the nodes are the nodes are allotted color 3 when i is odd and dominated by color class 1. Then for ⌊ ⌋ , the ;

color 2 when i is even and the nodes ; are painted with nodes ; ; are dominated by color class color respectively. The nodes ; , ; and are respectively. The nodes ; and are dominated by color allotted color 3, ⌈ ⌉ and 2 respectively. class ⌈ ⌉

The node w and for , the nodes are Every neighboring node is given different color and also dominated by color class 1. Then for ⌈ ⌉ , the it is observed that every node of the graph dominates all the nodes of atleast one color class. Thus it is a dominator nodes ; ; are dominated by color class coloring of nodes and the dominator coloring number of respectively. Every neighboring node is given different color and also closed Helm graph where is given by it is observed that every node of the graph dominates all the nodes of atleast one color class. Thus it is a dominator ⌈ ⌉ . coloring of nodes and the dominator coloring number of Hence the dominator coloring number of closed Helm closed Helm graph where is graph where is given given by

by ⌈ ⌉ . ⌈ ⌉ .

C.Theorem : Case 3: When is odd and Every closed Helm graph denoted by where and n is odd, has dominator chromatic number For , the nodes are allotted color 2 when i is odd and color 3 when i is even. The nodes and w is

⌈ ⌉ painted with color 4 and 1 respectively. For ⌊ ⌋ the

{ . ⌈ ⌉ nodes ; are allotted color 3 when i is odd and color 2 when i is even and the nodes are allotted color Proof: ; Let the node set and edge set of the closed Sun graph respectively. The nodes ; and are painted with color

be 4 and ⌈ ⌉ respectively.

{ } { } { } { } { }. The procedure below explains the dominator coloring of nodes.

Case 1: When is even and

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Dominator Coloring Of Certain Graphs

The node w and for , the nodes are The domination number of closed helm graph denoted by dominated by color class 1. Then for ⌈ ⌉ , the where :, is given by nodes ; ; are dominated by color class respectively. The nodes and are dominated by ⌈ ⌉ ; { color class ⌈ ⌉ . ⌈ ⌉

Every neighboring node is given different color and also it is observed that every node of the graph dominates all the nodes of atleast one color class. Thus it is a dominator E. Corollary coloring of nodes and the dominator coloring number of Every closed helm graph denoted by where :, closed Helm graph where is satisfies the relation given by

⌈ ⌉ . {

Hence the dominator coloring number of closed Helm graph where is given by Proof: By applying proposition 3.1, theorem 3.2, 3.3 and

⌈ ⌉ lemma 3.4 we get the result. { .

⌈ ⌉

IV. DOMINATOR CHROMATIC NUMBER OF D. Lemma: FLOWER SNARK The domination number of closed helm graph denoted A. Proposition: by where :, is given by The chromatic number of Flower Snark Jn where n is odd and , is ⌈ ⌉ .

{ ⌈ ⌉ B.Theorem: Proof: Every Flower snark Jn when n is odd has dominator The closed helm graph denoted by has its node set chromatic number { } { }. . Case 1: Let the dominating set of the closed helm graph be Proof: The n copies of a star graph on 4 nodes are taken. The

{ ⌈ ⌉}. ; central node of each star is denoted by ai and the outer nodes Clearly every vertex in is adjacent to atleast one by bi, ci and di. Thus the node set of a flower snark is given node of DS and the dominating set DS has the minimum by cardinality. Hence the domination number of the closed { }. helm graph denoted by is given by The 6n edges of the flower snark are as follows: There are n copies of 3 edges { } which

⌈ ⌉ . gives 3n edges. Then a cycle of length n is constructed by

connecting the n nodes b1,... bn which adds n edges. Finally a cycle of length 2n is constructed by connecting the 2n Case 2: nodes c1, c2...,cn, d1, d2... dn which adds another 2n edges. Let the dominating set of the closed helm graph be The following procedure gives dominator coloring of nodes

{ ⌊ ⌋} { }. ; For , the nodes are allotted color i+3. Clearly every vertex in is adjacent to atleast one For , the nodes are painted with color 1 node of and and the dominating set DS has the minimum when i is odd and color 2 when i is even. The node is cardinality. Hence the domination number of the closed allotted color 3. For , the nodes are assigned helm graph denoted by is given by color 1 when i is odd and color 2 when i is even. For

⌈ ⌉ . , the nodes are allotted color 2 when i is odd and color 1 when i is even. Case 3: Then the nodes for are dominated Let the dominating set of the closed helm graph be by color class respectively. { 4}. Clearly every vertex in is adjacent to atleast one node of and and the dominating set DS has the minimum cardinality. Hence the domination number of the closed helm graph denoted by is given by

⌈ ⌉.

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Every neighbouring node is given different color and also V. DOMINATOR CHROMATIC NUMBER OF it is observed that every node of the graph dominates all the WATKINS SNARK nodes of atleast one color class. Thus it is a dominator coloring of nodes and the dominator coloring number of A.Proposition: flower snark Jn where n is odd and , is given by The chromatic number of Watkins snark is . . B.Theorem : The dominator chromatic number of Watkins Snark is

given by ⌈ ⌉. 푐

푑 Proof: 1 The number of nodes of Watkins Snark is 50 and the 2 푎 6 number of edges is 75. i.e., . The node 2 set and the edge set of Watkins snark is given by 1 푑 { }. 푐4

푏 { : } 7 푎 1 { 4 푎 ;9 ;5 ;8 ; ;6 ; ;4 푏 2 푏 } 4 3 5 푐 { ; : } { ;7 : 푑4 1 2 2 1 } { : } { ; } The following procedure gives dominator coloring of nodes 푏5 푏 For ⌊ ⌋ , the nodes are allotted color . ; 1 8 4 2 For ⌈ ⌉ , the nodes :9 : 9 are painted with 푎5 푎 8 푐5 푑 color , the nodes : 9 are allotted color respectively.

For , the nodes are painted with color . :5 2 1 For , the nodes are painted with color . 5 푑5 푐 For ⌈ ⌉ , the nodes

are allotted ; :⌊ ⌋ : : : : 9 :4

color respectively.

푣 푣 푣 1 3 2 Figure 2: Dominator chromatic number of Flower snark 4 1 푣 5 푣4 J5 is 8. i.e, 1 푣6 푣 1 8 푣 2 2 4 푣7 푣 푣 1 1 5 푣9 1 2 C.Lemma: 2 7 푣44 푣 푣48 2 푣 5 푣 6 푣 푣49 Every Flower snark Jn when n is odd has domination 4 1 푣 2 1 number 푣47 푣 2 1 7 . 2 푣 8 푣 45 푣 1 푣 4 2 46 푣 푣 6 Proof: 푣4 1 푣5 9 8 1 1 The node set of a flower snark is given by

{ }. 1 푣 9 1 푣 Let { }. Clearly every node in 2 1 푣 1 8 푣 9 1 푣 푣4 푣 has atleast a neighbor in and the dominating set DS has 7 7 푣 6 푣 4 2 1 푣 8 푣 the minimum cardinality. 1 푣 5 2 6 2 1 Hence the domination number of Flower Snark Jn when n is 푣 1 푣 5 푣 4 푣 1 odd is given by 푣 2 푣 2 9 푣 . 푣

D. Corollary: Figure 3: Dominator chromatic number of Watkins The Flower Snark J when n is odd satisfies the relation n snark is

. Proof: The result follows from proposition 4.1, theorem 4.2 and lemma 4.3.

Dominator Coloring Of Certain Graphs

For , the nodes ; ; are dominated { } { 5 : by color class and the nodes :6 are dominated by } { } ;4 : 4 color class respectively. For ⌊ ⌋, the nodes { 5 ; 4 5 : 7 :4 ; } { ;5 } :6 :7 :8 are dominated by color class respectively. For ⌈ ⌉ , the nodes The following procedure gives dominator coloring of nodes : 9 : : are dominated by color class , the For ⌈ ⌉, the nodes ; : 8 are alloted color nodes : 9 : : are dominated by color class 6 , the nodes : 9 :4 :4 are dominated by and the nodes ; are alloted respectively. color class , the nodes : are dominated by For ⌊ ⌋, the nodes are painted with color . 4 color class respectively. For the nodes are alloted Every neighbouring node is given different color and also 5 : : 4 : it is observed that every node of the graph dominates all the color respectively. The nodes : ; are nodes of atleast one color class. Thus it is a dominator painted with color respectively. coloring of nodes and the dominator coloring number of Watkins snark is given by 3 1 푣 푣 2 ⌈ ⌉ 푣 C.Lemma : 2 푣 5 The domination number of Watkins snark is given by 1 8 푣 6 푣 7 ⌈ ⌉ . 7 1 1 푣 8 2 푣 4 Proof: 푣 4 2 푣4 The node set of a Watkins snark is given by 푣 5 푣 푣 { }. 2 4 푣5 1 1 Let { ;9 ;7 ; ⌈ ⌉}. 푣 6 푣 Clearly every node in V – DS has atleast a neighbor in DS 1 푣6 and the dominating set DS has the minimum cardinality. 푣 7 Hence the domination number of Watkins snark is given by 2 10 1 ⌈ ⌉ . 푣 1

2 2 D. Corollary: 푣 9 푣 9 The Watkins snark satisfies the relation 1 푣 9 푣 8 1 푣 . 푣 2 푣 6 Proof: 푣7 By applying proposition 5.1, theorem 5.2 and lemma 5.3, 푣 1 the result follows. 1 푣8 푣9 2 5

VI. DOMINATOR CHROMATIC NUMBER OF Figure 4: Dominator chromatic number of Double star DOUBLE STAR SNARK snark is, A.Proposition:

Then for ⌈ ⌉ , the nodes are The chromatic number of double star snark is 6 ; ;

. dominated by color class respectively. For ,

B.Proposition : the nodes are dominated by color class : 5 The domination number of double star snark is and color class

. and the nodes : : 4 C.Theorem: are dominated by color class respectively. For

The dominator chromatic number of double star snark is the nodes are dominated by color 5 : : 4 given by class and the nodes are dominated by color : . class . Proof: The number of nodes of the double star snark is 30 and the number of edges is 45. i.e., .

The node set is given by { }. And the edge set is given by

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Every neighbouring node is given different color and also it is observed that every node of the graph dominates all the nodes of atleast one color class. Thus it is a dominator coloring of nodes and the dominator coloring number of Double star snark graph is

D. Result : The Double star Snark G satisfies the relation . Proof: By applying proposition 6.1, 6.2 and theorem 6.3 we get the result.

REFERENCES

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