B-Chromatic Number of &amp;lt;i&amp;gt;M&amp;lt;/i&amp;gt;[&amp;lt;i&amp;gt;C&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;],&amp;lt;i&amp;gt;M&amp;lt;/i&amp;gt;[&amp;lt;i&amp;gt;P&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;],&amp;lt;i&amp;gt;M&amp;lt;/i&amp;gt;[&amp;lt;i&amp;gt;F&amp;lt;sub&amp;gt;1,n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;] and &amp;lt;i&amp;gt;M&amp;lt;/i&amp;gt;[&amp;lt;i&amp;gt;W&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;

Duraisamy Vijayalakshmi,Narayanan Roopesh,Kandasamy Thilagavathi

doi:10.4236/ojdm.2011.12010

Abstract

In this paper, we discuss about the b-colouring and b-chromatic number for middle graph of Cycle, Path, Fan graph and Wheel graph denoted as M[Cn],M[Pn],M[F1,n] and M[Wn] .

Highlights

Let G be a finite undirected graph with no loops and multiple edges
We discuss about the b-colouring and b-chromatic number for middle graph of Cycle, Path, Fan graph and Wheel graph denoted as M Cn, M Pn, M F1,n and M Wn
A coloring is called a b-coloring [1], if for each color i there exists a vertex xi of color i such that every color j ≠ i, there exists a vertex yj of color j adjacent to xi, such a vertex xi is called a dominating vertex for the colour class i or color dominating vertex which is known as b-chromatic vertex

Summary

Introduction

Let G be a finite undirected graph with no loops and multiple edges. A coloring (i.e., proper coloring) of a graph G = (V,E) is an assignment of colors to the vertices of G, such that any two adjacent vertices have different colors. The b-chromatic number of a graph G, denoted by (G) is the largest positive integer k such that G has a b-colouring by k colors. The b-chromatic number of a graph was introduced by R.W. Irwing and manlove [2] in the year 1999 by considering proper colorings that are minimal with respect to a partial order defined on the set of all partitions of V(G). Irwing and manlove [2] in the year 1999 by considering proper colorings that are minimal with respect to a partial order defined on the set of all partitions of V(G) They proved that determining (G)[3] is NP-hard for general graphs, but polynomial for trees. Let G be a graph with vertex set V(G) and the edge set E(G). Y in the vertex set of M(G) are adjacent in M(G) in case one of the following holds; 1) x, y are in E(G) and x, y are adjacent in G 2) x is in V(G), y is in E(G), and x, y are incident in G

Theorem