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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. Published By: Blue Eyes Intelligence Engineering Retrieval Number: B10611282S18/18©BEIESP 262 & Sciences Publication 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 5 2 nodes. 푣5 푣 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 ⌈ ⌉. Published By: Blue Eyes Intelligence Engineering Retrieval Number: B10611282S18/18©BEIESP 263 & Sciences Publication International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-2S, December 2018 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 ⌈ ⌉ .
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