If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" is said to be equitably k -colorable. The equitable chromatic number ?(G) is the smallest integer k such that G is equitably k-colorable. The notion of an equitable coloring was ?rst introduced by W. Meyer. The Equitable Coloring Conjecture (E CC) : Let be a connected graph. If G is neither a complete graph nor an odd cycle , then . The Equitable ? -Coloring Conjecture (E? CC) : A connected graph is equitably ? -colorable if G is different from , and for all . It is also immediate to see that the E ? CC implies the ECC. In C hapter 2 positive evidence for the important equitable ?- coloring conjecture is supplied from graph justify; MARGIN: 0cm 0cm 0pt" In Chapter 3 a ppropriate conjectures for equitable ? -coloring of disconnected graphs are studied. In Chapter 4 the equitable ?-coloring conjecture for planar graphs are considered . It is proved that each planar graph in various justify; MARGIN: 0cm 0cm 0pt" In Chapter 5 the equitable ?-coloring conjecture for outerplanar graphs are studied. It is proved that every outerplanar graph with maximum degree at most ? admits an equitable k-coloring for every .