  We prove that the motion of a single hole induces $SU(N)$ flavor-singlets in
the $U=\infty$ $SU(N)$ (Fermi) Hubbard model on a Husimi-like tree graph. The
result is also generalized to certain $t$-$J$ models with antiferromagnetic
interactions and singlet hopping terms typically neglected in the literature.
This is an $SU(N)$ generalization of the "counter-Nagaoka theorem" introduced
in [Phys. Rev. B 107, L140401 (2023)]. Our results suggest the existence of
antiferromagnetic or resonating-valence-bond (RVB)-like polarons in the $t$-$J$
models on a more realistic non-bipartite lattice. Such antiferromagnetic/RVB
polarons may be relevant for a novel strong-coupling mechanism of
superconductivity or other exotic fractionalized phases of matter.
