Monochromatic Disconnection Of Graphs

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. For a connected graph G, the monochromatic disconnection number of G, denoted by md(G), is the maximum number of colors that are needed in order to make G monochromatically disconnected. We show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type results for md(G). (C) 2020 Elsevier B.V. All rights reserved.