 One-loop cubic-quartic scalar vertex classification

For a connected real-scalar Feynman diagram, degree counting and $L=I-V+1$ give $V_3+2V_4=E+2L-2$. A one-loop one-particle-irreducible Feynman diagram is a cycle: every vertex has two internal half-edges. For three labelled external legs, the possibilities are one cubic triangle with symmetry factor one and three mixed cubic-quartic bubbles with symmetry factor two. For four labelled external legs, there are three quartic bubbles with symmetry factor two, six mixed triangles with symmetry factor one, and three cubic boxes with symmetry factor one. Cyclic ordering modulo reversal explains the three boxes.