Exact hole-induced resonating-valence-bond ground state in certain U=∞ Hubbard models

We prove that the motion of a single hole induces the nearest-neighbor resonating-valence-bond ground state in the $U=\ensuremath{\infty}$ Hubbard model on a triangular cactus---a treelike variant of a kagome lattice. The result can be easily generalized to $t\ensuremath{-}J$ models with antiferromagnetic interactions $J\ensuremath{\ge}0$ on the same graphs. This is a weak converse of Nagaoka's theorem of ferromagnetism on a bipartite lattice.