On b-chromatic number of sun let graph and wheel graph families

Abstract

A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Concepts and questions of graph coloring arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoretical Computer Science. A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color class. The b-chromatic number of a graph G, denoted by uðGÞ, is the maximal integer k such that G may have a b-coloring with k colors. In this paper, we obtain the b-chromatic number for the sun let graph S n , line graph of sun let graph LðS n Þ, middle graph of sun let graph MðS n Þ, total graph of sun let graph TðS n Þ, middle graph of wheel graph MðW n Þ and the total graph of wheel graph TðW n Þ. 1991 MATHEMATICS SUBJECT CLASSIFICATION: 05C15; 05C75; 05C76 ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

On (1,2)- Blast Domination Number for Some Total Graphs

“…A non-empty subset V D  is called a (1,2)dominating set in a graph having the property that for every vertex in there is atleast one vertex in at distance 1 from and a second vertex in at distance On (1,2)-Blast Domination Number for Some Total Graphs P.Rajakumari, K. Ameenal Bibi  atmost 2 from .The order of the smallest (1,2)-dominating set of is called the (1,2)-domination number of denoted by Definition 2.10 [12] Let be a graph with vertex set and edge set The total graph [3] of , denoted by is defined in the following way. The vertex set of is and Two vertices of are adjacent in , if either (i) in and is adjacent to in (or) (ii) in and are adjacent in (or) (iii) is in is in and are incident in…”

On b-coloring of central graph of some graphs

“…A Triangular Snake [8] is obtained from a path x 1 ; x 2 ; : : : ; x n by joining x i and x i+1 to a new vertex y i for 1 i n. That is, every edge of a path is replaced by a triangle C 3 . The n-Sunlet graph S n is a graph [11] with cycle C n and each vertex of the cycle attached to one pendent vertex. Each n-sunlet graph consists 2n nodes and 2n edges.…”

“…Each n-sunlet graph consists 2n nodes and 2n edges. A Helm H n , n 3 is the graph [11] obtained from the Wheel W n by adding a pendent edge at each vertex on the rim of the Wheel W n . The Central graph [9] of G, denoted by C (G) is obtained by subdividing each edge of G exactly once and joining all the non-adjacent vertices of G in C (G).…”

“…. x n by joining x i and x i+1 to two new vertices y i and z i for i = 1; 2; : : : n. A Gear graph [11] is obtained from the Wheel W n by adding a vertex between every pair of adjacent vertices of rim of the Wheel W n . A Closed Helm CH n is the graph obtained by taking a Helm H n and adding edges between the pendent vertices.…”

The b-chromatic number of Mycielskian of some graphs

“…In 2014, Vivin and Venkatachalam (2014a) obtained the b-chromatic number of middle and total graph of fan graph. In 2014, Vivin and Venkatachalam (2014b) discussed the b-chromatic number for the sun let graph S n , line graph of sun let graph L(S n ), middle graph of sun let graph M(S n ), total graph of sun let graph T(S n ), middle graph of wheel graph M(W n ) and the total graph of wheel graph T(W n ). In 2012, Vijayalakshmi and Thilagavathi (2012b) obtained the b-chromatic number of corona product of path, cycle and star graph with complete graph, the strong product of path with cycle and Cartesian product of cycles.…”