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Graph-theoretic analysis uncovers exact weak saturation numbers for three-dimensional cubes in complete graphs, disproving a prior conjecture and indicating tighter asymptotic bounds.
We study the weak saturation number of the three-dimensional cube Q3 in complete graphs. By normalizing the first activated cube and performing complete finite enumerations, we determine wsat(K8,Q3)=15, wsat(K9,Q3)=16, and wsat(K10,Q3)=18. The order-nine value corrects a reported value of 17 and disproves the associated proposal wsat(K_n,Q3)=2n-1. We also prove a complete-anchor gluing lemma and give a certified 28-edge graph on 11 vertices with a complete seven-vertex anchor. This yields wsat(K_(N+4),Q3)<=wsat(K_N,Q3)+7 and therefore wsat(K_n,Q3)<=floor((7n+2)/4) for every n>=9. Combined with a prior lower bound, the asymptotic constant lies between 11/7 and 7/4. Status: Public Beta; internally verified candidate proof; external mathematical review pending.
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