Tight Bounds on Probabilistic Zero Forcing on Hypercubes and Grids

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.

Sharp thresholds in adaptive random graph processes

“…We conclude with the observation that a sharp threshold does not always exist for the semi‐random process, which follows as a byproduct of the result in [2].…”

“…Given a list of fixed graphs, none of which are forests, we prove that a sharp threshold does not exist for the property of containing at least one of these fixed graphs. The results in this section follow from the below result of Behague et al [2], and we include the proofs for completeness.…”

“…Fix a graph $$$ H $$$ of $$$ L $$$ of degeneracy $$$ d $$$. We first describe the strategy from [2]. Since $$$ H $$$ is $$$ d $$$‐degenerate, we may consider an ordering of the vertices of $$$ H $$$ (say, $$$ \left({v}_1,\dots, {v}_k\right) $$$) such that $$$ {v}_i $$$ has at most $$$ d $$$ neighbours in $$$ \left\{{v}_1,\dots, {v}_{i-1}\right\} $$$.…”

“…The semi‐random graph process was suggested by Peleg Michaeli, introduced formally in [5], and studied in [2–4,7,13–15]. The process is a one player game in which the player begins with the empty graph on $$$ \left[n\right] $$$.…”