 Solution

There are three vertex types, with all legs labeled in the following original drawings:

The gluon cubic interaction vertex follows from the non-Abelian gauge field strength of Quantum chromodynamics. The electroweak Yang-Mills theory Lagrangian, after expressing the fields in the physical charged and neutral basis, produces the two charged-W boson vertices. In particular there are no fundamental $\gamma \gamma \gamma$, $ZZZ$, $ZZ\gamma$, or $Z\gamma \gamma$ vertices, and the separate colour and electroweak gauge factors do not generate a mixed $gg\gamma$ or $ggZ$ vertex. The count is by the boson labels requested, not by separate assignments of gluon colour indices.