Bounds on expected propagation time of probabilistic zero forcing

Abstract

Probabilistic zero forcing is a coloring game played on a graph where the goal is to color every vertex blue starting with an initial blue vertex set. As long as the graph G is connected, if at least 1 vertex is blue then eventually all of the vertices will be colored blue. The most studied parameter in probabilistic zero forcing is the expected propagation time ept(G). We significantly improve on upper bounds for ept(G) by Geneson and Hogben and by Chan et al. in terms of a graph’s order and radius. We prove the bound ept(G)=Orlognr. We also show using Doob’s Optional Stopping Theorem that ept(G)≤n2+O(logn). Finally, we derive an explicit lower bound ept(G)≥log2log2(2n).