Probabilistic Zero Forcing with Vertex Reversion

TL;DR

This paper introduces reversion probabilistic zero forcing (RPZF), a new graph coloring process where vertices can revert from blue to white, and provides tools to analyze the probabilities of complete coloring states.

Contribution

The paper presents RPZF, extending probabilistic zero forcing by allowing vertex reversion, and develops methods to compute transition probabilities and thresholds for graph coloring.

Findings

RPZF models vertex reversion in probabilistic zero forcing.

Tools for calculating transition probabilities and expected times are established.

Thresholds for complete coloring in specific graph families are derived.

Abstract

Probabilistic zero forcing is a graph coloring process in which blue vertices "infect" (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. We introduce reversion probabilistic zero forcing (RPZF), which shares the same infection dynamics but also allows for blue vertices to revert to being white in each round. We establish a tool which, given a graph's RPZF Markov transition matrix, calculates the probability that the graph turns all white or all blue as well as the time at which this is expected to occur. For specific graph families we produce a threshold number of blue vertices for the graph to become entirely blue in the next round with high probability.