The chromatic number of a graph G , denoted by ?(G) , is the minimum number of colors required to color the vertices of G such that adjacent vertices receive different colors. In recent decades, the concepts of chromatic number, independence number and clique number have been studied widely in the literature. Since the computation of ?(G) is NP-hard, finding bounds for ?(G) in terms of various parameters of the given graph $ G $, such as clique number ? (G) and the maximum degree ? (G) has received much attention. I n this thesis we are going to investigate research studies on a well-known conjecture, called Reed's Conjecture. Let G be a graph of order n with chromatic number ?(G) , maximum degree ? (G) and clique number ? (G). We know that, for every graph G , ?(G)??(G)??(G)+1 . In 1941 [3] Brooks proved that if a connected graph G is neither an odd cycle nor a complete graph, then ?(G)?? (G) . This theorem says that if G is a graph with ? (G)?3 and ? (G)?? (G) , then ?(G)?? (G) . In [26] Reed conjectured that, ?(G) is bounded above by convex combinations of the trivial lower bound ? (G) and the trivial upper bound ? (G)+1 .
