Colorful monochromatic connectivity

An edge-coloring of a connected graph is monochromatically-connecting if there is a monochromatic path joining any two vertices. How “colorful” can a monochromatically-connecting coloring be? Let m c ( G ) denote the maximum number of colors used in a monochromatically-connecting coloring of a graph G . We prove some nontrivial upper and lower bounds for m c ( G ) and relate it to other graph parameters such as the chromatic number, the connectivity, the maximum degree, and the diameter.