Let \(G=(V,E)\) be a finite connected graph along with a coloring of the vertices of G using the colors in a given set X. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing procedure in terms of the number of vertices in the graph on which the procedure is being applied. We then focus on multi-color forcing with three colors and analyze the end states of certain families of graphs, including complete graphs, complete bipartite graphs, and paths, based on various initial colorings. We end with a few directions for future research.

令\（G =（V，E）\）是一个有限的连通图，并使用给定集合X中的颜色对G的顶点进行着色。在本文中，我们介绍了多色强制，即图上的零强制的一般化，并给出了无论使用多少颜色，多色强制过程终止的条件。根据应用该过程的图形中的顶点数，我们给出了终止强制过程所需的步骤数的上限。然后，我们将重点放在具有三种颜色的多色强制上，并基于各种初始着色，分析某些图族的最终状态，包括完整图，完整二部图和路径。我们以一些未来的研究方向作为结尾。