Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between $P_3$, the probability that all three vertices in the triangle connect, and $P_0$, the probability that none connect. A linear approximation for $P_3\left(P_0\right)$ is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for $P_3\left(P_0\right)$ gives thresholds for the kagomé, site-bond honeycomb, $\left(3\text{−}{12}^2\right)$ lattice, and “stack-of-triangle” lattices that compare favorably with numerical results.

• Received 3 December 2008

DOI:https://doi.org/10.1103/PhysRevE.79.020102