Multi-color forcing in graphs

TL;DR

This paper introduces multi-color forcing, a generalization of zero-forcing on graphs, providing conditions for termination, bounds on steps, and analyzing end states for specific graph families with three colors.

Contribution

It extends zero-forcing to multi-color scenarios, establishes termination conditions, and analyzes end states for key graph families with three colors.

Findings

Multi-color forcing terminates under certain conditions.

An upper bound on the number of steps for termination.

End states vary for different graph families with three colors.

Abstract

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.