= Triangle replacement for self-avoiding walks
{title2=$y^2=x^3+x^4$}
Replace every degree-three <vertex of a graph> in one class of a <bipartite graph> by a triangle with one port for each incident <edge>. A <self-avoiding walk> whose endpoints remain in the other class cannot make two completed passages through the same triangle: each passage needs two unused ports and only three exist. Each old two-edge passage has exactly two replacements, of lengths three and four. Therefore the restricted <generating functions> satisfy $Z_H^0(x)=Z_G^0(\sqrt{x^3+x^4})$. If the old radius is $1/\mu$, the new radius obeys $\rho^3+\rho^4=\mu^{-2}$.
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