Bounds on expected propagation time of probabilistic zero forcing

Shyam Narayanan, Alec Sun

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<math><mi is="true">G</mi></math> 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)<math><mrow is="true"><mtext is="true">ept</mtext><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math>. We significantly improve on upper bounds for ept(G)<math><mrow is="true"><mtext is="true">ept</mtext><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math> 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.<math><mrow is="true"><mtext is="true">ept</mtext><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">O</mi><mfenced close=")" open="(" is="true"><mrow is="true"><mi is="true">r</mi><mo class="qopname" is="true">log</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">r</mi></mrow></mfrac></mrow></mfenced><mo is="true">.</mo></mrow></math> We also show using Doob’s Optional Stopping Theorem that ept(G)≤n2+O(logn).<math><mrow is="true"><mtext is="true">ept</mtext><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mo class="qopname" is="true">log</mo><mi is="true">n</mi><mo is="true">)</mo></mrow><mo is="true">.</mo></mrow></math> Finally, we derive an explicit lower bound ept(G)≥log2log2(2n).<math><mrow is="true"><mtext is="true">ept</mtext><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><msub is="true"><mrow is="true"><mo class="qopname" is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><msub is="true"><mrow is="true"><mo class="qopname" is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mrow is="true"><mo is="true">(</mo><mn is="true">2</mn><mi is="true">n</mi><mo is="true">)</mo></mrow><mo is="true">.</mo></mrow></math>