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