  We study tensor models based on $O(N)^r$ symmetry groups constructed out of
rank-$r$ tensors with order-$q$ interaction vertices. We refer to those tensor
models for which $r<q-1$ as \textit{subchromatic}. We focus most of our
attention on sextic ($q=6$) models with maximally-single-trace interactions. We
show that only three subchromatic sextic maximally-single-trace interaction
vertices exist: these are the $r=3$ prism, the $r=3$ wheel (or $K_{3,3}$) and
the $r=4$ octahedron. For theories based on these interactions we demonstrate
that the set of Feynman diagrams that contribute to the free energy in the
large $N$ limit are melonic (or closely related to melonic diagrams, in the
case of the prism) and thus can be explicitly summed.
