Monochromatic Vertex-Disconnection Colorings of Graphs

Let G be a vertex-colored connected graph. A subset U of the vertex set of G is called monochromatic , if all vertices of U are assigned the same color. The vertex-colored graph G is called monochromatic vertex-disconnected if for any two distinct vertices x and y , there is a monochromatic vertex-subset S of G such that x and y belong to different components of G - S if x and y are nonadjacent, and if x and y are adjacent, then x or y has the same color as S and x and y belong to distinct components of ( G - x y ) - S . The monochromatic vertex-disconnection number of a connected graph G , denoted by mvd ( G ) , is defined as the maximum number of colors that are allowed to make G monochromatic vertex-disconnected. The concept is inspired by the concepts of rainbow vertex-disconnection number rvd ( G ) and monochromatic disconnection number md ( G ) . In this paper, we present some sufficient conditions for a connected graph G to have mvd ( G ) = 1 and show that almost all graphs have monochromatic vertex-disconnection number 1. Moreover, we present Nordhaus–Gaddum-type results for the new parameter mvd ( G ) . At last, we investigate the monochromatic vertex-disconnection numbers for four graph products.