Strictly balanced graph ($e(F)/v(F)<e(H)/v(H)$)

A finite graph $H$ with at least one edge is strictly balanced when every proper nonempty subgraph $F$ has smaller edge-to-vertex ratio. It is connected and has no isolated vertices: otherwise a component of at least the average density, or deletion of isolated vertices, violates the inequality. The density condition suppresses overlapping copies at the sparse appearance scale.

Here $v,e,a$ are the numbers of vertices, edges and graph automorphisms of $H$. Vertex-disjoint ordered $r$-tuples contribute $(n{)}_{rv}{p}^{re}/a^r\to (c^e/a{)}^r$ to the factorial moment. Adding a copy overlapping the previous union in $f$ vertices and $g$ edges changes the power of $n$ by $-f+(v/e)g\le 0$, strictly when the intersection is a proper nonempty subgraph. The first overlap is strict because $H$ is connected and the previous copies are disjoint. Finitely many overlap types therefore contribute $o(1)$. The factorial-moment criterion for Poisson convergence gives the result.