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 $md(G)$.

中文翻译：

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图的单色断开

对于边着色图$G$，如果$M$ 的边被着色为相同颜色，我们称$G$ 的边切割$M$ 为单色。如果 $G$ 的任何两个不同的顶点被单色边缘切割分开，则图 $G$ 被称为单色断开。对于连通图$G$，$G$的单色断开数，用$md(G)$表示，是为了使$G$单色断开所需的最大颜色数。我们将证明几乎所有图都有等于 1 的单色断开数。我们还获得了 $md(G)$ 的 Nordhaus-Gaddum 型结果。