Monochromatic disconnection of graphs

TL;DR

This paper introduces the concept of monochromatic disconnection in graphs, defining the monochromatic disconnection number, and explores its properties, including that most graphs have a disconnection number of 1 and establishing Nordhaus-Gaddum-type bounds.

Contribution

It defines the monochromatic disconnection number for graphs and analyzes its typical value and bounds, providing new insights into graph coloring and connectivity.

Findings

Almost all graphs have md(G) = 1.

Established Nordhaus-Gaddum-type results for md(G).

Provided theoretical bounds and properties of md(G).

Abstract

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a 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, denoted by $md(G)$, of $G$ is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. We will show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type results for $md(G)$.