Exact Hole-induced Resonating-Valence-Bond Ground State in Certain $U=\mathrm{\infty }$ Hubbard Models

We prove that the motion of a single hole induces the nearest-neighbor resonating-valence-bond ground state in the $U=\mathrm{\infty }$ 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 $J\ge 0$ on the same graphs. This is a weak converse of Nagaoka’s theorem of ferromagnetism on a bipartite lattice.