Tight Bounds on the Probabilistic Zero Forcing on Hypercubes and Grids

TL;DR

This paper investigates the probabilistic zero forcing process on hypercubes and grids, providing tight bounds on the propagation time and analyzing how randomness affects the spread of the blue color in these graph structures.

Contribution

It introduces tight bounds on the propagation time of probabilistic zero forcing specifically on hypercubes and grids, advancing understanding of stochastic graph coloring processes.

Findings

Derived tight bounds for hypercubes

Established bounds for grid structures

Analyzed the impact of probability on forcing time

Abstract

Zero forcing is a deterministic iterative graph colouring process in which vertices are coloured either blue or white, and in every round, any blue vertices that have a single white neighbour force these white vertices to become blue. Here we study probabilistic zero forcing, where blue vertices have a non-zero probability of forcing each white neighbour to become blue. We explore the propagation time for probabilistic zero forcing on hypercubes and grids.