A Vertex- Coloring of a Graph Is Called a Bounded Monochromatic Component Colorings (Bmc) If Every Color-Class Induces Monochromatic Components Containing at Most a Certain Bounded Num¬Ber of Vertices. a Proper Coloring For Instance Is a Bmc Coloring In Which Every Color-Class Induces Monochromatic Components of Order One. We Investigate Three Different Aspects of Bmc Colorings.