= One-loop cubic-quartic scalar vertex classification
{title2=$V_3+2V_4=E\quad(L=1)$}
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.
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