Chromatic polynomial inclusion-exclusion formula ($p_G(x)={\sum }_{F\subseteq E(G)}(-1{)}^{|F|}{x}^{c(F)}$)

Here $c(F)$ is the number of connected components of a graph in the spanning subgraph $(V(G),F)$, including isolated vertices. Apply the inclusion-exclusion principle to the conditions that the endpoints of an edge have equal colours. Imposing all such conditions for $F$ forces a single colour on each connected component of a graph, yielding $x^{c(F)}$ for positive integer $x$. The equality extends to a polynomial identity.