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. The monochromatic disconnection number, denoted by md(G), of a connected graph G is the maximum number of colors that are allowed to make G monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

中文翻译：

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单色断开：Erdős-Gallai 型问题和产品图

对于边缘彩色图形ģ，我们称之为边缘切割中号的ģ单色如果边缘中号与一个相同的颜色着色。该曲线图G ^是如果任何两个不同的顶点称为单色断开ģ由单色边缘切割分离。连接图G的单色断开数，用md ( G )表示，是允许使G的最大颜色数单色断开。在本文中，我们解决了单色断开的 Erdős-Gallai 型问题，并给出了四种图积的单色断开数，即笛卡尔积、强积、字典积和张量积。