ACYCLIC COLORING ON TRIPLE STAR GRAPH FAMILIES

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1 ACYCLIC COLORING ON TRIPLE STAR GRAPH FAMILIES Abstract 1 D.Vijayalakshmi Assistant Professor Department of Mathematics Kongunadu Arts and Science College Coimbatore 2 K. Selvamani, Research Scholars Department of Mathematics Kongunadu Arts and Science College Coimbatore selvamathz94@gmail.com An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The acyclic chromatic number a(g) of a graph G is the least number of colors needed in any acyclic coloring of G. In this paper, we obtain acyclic chromatic number of central graph, middle graph, total graph and line graph of triple star graph, denoted by,, and ( ) respectively. Mathematics Subject Classification: 05C15 Keywords: Acyclic chromatic number, central graph, middle graph, total graph, line graph. 1. Introduction and Preliminaries All graphs considered here are finite, undirected, having no loops and multiple edges. Let G be a graph with vertex set V(G) and edge set E(G). A proper coloring of the vertices of a graph G is an assignment of colors to the vertices so that no two neighbors get the same color. A vertex coloring of a graph G = (V, E) is an acyclic if it is a proper coloring and there is no cycle in the subgraph induced by the vertices of any two of the colors. An acyclic chromatic number of G, denoted by a(g), it the least number of colors in an acyclic coloring of G. A star coloring of a graph is a proper coloring such that the subgraph induced by the union of any two color classes is a disjoint collection of stars. The acyclic and star chromatic numbers of G are define analogously to the chromatic number and are denoted by (G) and (G) respectively. The concept of acyclic coloring of a graph was introduced by Grunbaum [5] determining acyclic coloring graph G is a hard problem, Kostochka [7] proved that the problem is an NP-complete to decide an arbitrary graph G. Borodin et al. [2, 3] concentrated on the family of planar graphs with large 1-planar graphs and girth, respectively. Sopena [9] and Fertin et al. [4] consider the family of outer graphs 1 396

2 and d-dimensional grids respectively. [5] considered the family of graphs with maximum degree 3 and Skularattanakulchari [10] consider the family of graphs with maximum degree 4. These are the computational difficultly involved in determining acyclic coloring different families of graphs. Andrew Lyons [1] found every acyclic coloring of a cographs is also a star coloring and he gave a linear time algorithm finding an optimal acyclic and star coloring of a cograph. Kishore Yadav et al. [8] found the acyclic coloring the family of graph with maximum degree 5 as 8 and also they gave a linear time algorithm to achieve this bound. Let be a triple star graph obtained from by adding new pendant vertices,, of the existing pendant vertices. The central graph [11] C(G) of a graph G is med by adding an extra vertex on each edge of G, and then joining each pair of vertices of the original graph which were previously non-adjacent. The middle graph [6] of G, denoted by M(G) is defined as follows. The vertex set of M(G) is V(G) U E(G). Two vertices x,y in the vertex set of M(G) are adjacent in M(G) in case one of the following holds: i) x,y are in E(G) and x,y are adjacent in G. ii) x is in V(G), y is in E(G), and x,y are incident in G. The total graph [11] denoted by T(G) has the vertex set as union of V(G) and E(G), and the edges connecting all elements of this vertex set which are incident or adjacent in G. The line graph [6] of G denoted by L(G) is a graph whose vertex set corresponds to E(G) and two vertices in L(G) are adjacent if and only if the corresponding edges in G are adjacent. 2. Acyclic coloring on central graph of triple star graph Theorem: 2.1 For any triple star graph, ( ) = 2. Proof Let ( ) and the number of edges in ( ) is. By the definition of central graph, the each edge can be subdivided by the vertices : 1, : 1, : 1 respectively. The vertex set of ( )is ( ) =. Clearly, the vertices ms a clique of order and the vertices ms a clique of order. There ( )

3 Algorithm: 2.1 end end end end By above the algorithm 2.1, it is clear that every cycle on four vertices contains atleast 3 distinct colors. Thus, [ ( )]. Hence, ( ). 3. Acyclic coloring on middle graph of triple star graph Theorem: 3.1 For any triple star graph, ( ) = Proof Let ( ) and the number of edges in ( ) is. By the definition of middle graph, let be the newly introduced vertices in the edge joining of, and respectively. The vertex set of ( )is 3 398

4 ( ) = Clearly, the vertices ms a clique of order. For, the vertex is adjacent with and. Theree [ ( )]. Algorithm:3.1 end end end By the above algorithm 3.1, any cycle on four vertices uses at least three distinct colors. Thus, [ ( )]. Hence, ( ). 4. Acyclic coloring on total graph of triple star graph Theorem: 4.1 For any triple star graph, ( ) =. Proof Let ( ) and the number of edges in ( ) is. By the definition of total graph, let be the newly introduced vertices in the edge joining of respectively. The vertex set of ( )is V[ ( ) = 4 399

5 Clearly, the vertices ms a clique of order. For the vertex is adjacent to and is adjacent with the vertices and, also the vertex is adjacent with and. Theree [ ( )]. Algorithm: 4.1 end end end By above algorithm 4.1, any cycle on three vertices uses at least three distinct colors. Thus, [ ( )]. Hence, ( ). 5. Acyclic coloring on line graph of triple star graph Theorem: 5.1 For any triple star graph, ( ) =. Proof: Let V( ) = and the number of edges in is. By the definition of line graph, each edge of can be taken as vertex in ( ). The vertex set of ( ) is defined as, ( [( )]) Clearly, the vertices induces a clique of order. Theree [ ( )]. Algorithm:

6 end end By above algorithm 5.1, it is clear that no cycle on 4 vertices is bicolored. Thus, [ ( )]. Hence, ( ). References: [1] Andrew Lyons, Acyclic and Star Colorings of Cographs, Computation Institute, University of Chicago and Mathematics and Computer Science Division, Argonne National Laboratory. [2] O.V. Borodin, A.V. Kostochka, A. Raspaud, E. Sopena, Acyclic coloring of 1-planar graphs, Diskretn. Anal. Issled. Oper. Ser. 1 6 (4) (1999) [3] O.V. Borodin, A.V. Kostochka, D.R. Woodwall, Acyclic coloring of planar graphs with large girth, J. Lond. Math. Soc. 60 (2) (1999) [4] G. Fertin, E. Godard, A. Raspaud, Acyclic and k-distance coloring of the grid, Inm. Process. Lett. 87 (1) (2003) [5] B. Grunbaum, Acyclic colorings of plannar graphs, Israel J. Math. 14(1973) [6] F. Harary, Graph theorey, Narosa Publishing Home, New Delhi [7] A.V. Kostochka, Upper bounds on the chromatic functions of graphs, Ph.D. Thesis, Novosibirsk, Russia, [8] Kishore Yadav, Satish Varagani, Kishore Kothapalli, V.Ch. Venkaiah, Acyclic vertex coloring of graphs of maximum degree 5, Discrete Mathematics 311(2011) [9] E. Sopena, The chromatic number of oriented graphs, Math. Notes 25 (1997) [10] S. Skulrattanakulchai, Acyclic colorings of subcubic graphs, Inm. Process. Lett. 92 (2004) [11] D. Vijayalakshmi, Study on b-chromatic coloring of graphs, Ph.D thesis, Bharathiar University, Coimbatore