The initial-vertex conjecture for randomized zero forcing on complete bipartite graphs

Let Ka,bK_{a,b} be a bidirectional complete bipartite graph with all edges of equal weight, where a>ba>b. Let AA and BB be its partite sets, with A=a|A|=a and B=b|B|=b. For vertices uAu\in A and vBv\in B, write eptrzf(Ka,b,x)\operatorname{ept_{rzf}}(K_{a,b},\\{x\\}) for the expected randomized zero-forcing propagation time when xx is the initial blue vertex. Initial-vertex conjecture.

eptrzf(Ka,b,v)<eptrzf(Ka,b,u).\operatorname{ept_{rzf}}(K_{a,b},\\{v\\})<\operatorname{ept_{rzf}}(K_{a,b},\\{u\\}).

This claims that, when the two partite sets have unequal sizes, starting in the smaller part gives strictly smaller expected propagation time than starting in the larger part. The surrounding results establish order-of-growth bounds for complete bipartite graphs, but do not resolve this strict comparison.

Sources & referencesView supporting material

Primary source

Jesse Geneson, Illya Hicks, Noah Lichtenberg, Alvin Moon and Nicolas Robles, “Randomized Zero Forcing”, arXiv:2602.16300 (2026).

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