Let $P_{H}(G, t)$ be the number of vertex colorings of a graph $G$ in $t$ colors that avoid having a monochromatic subgraph $H$. In particular, for $H$ given by a single edge we recover the usual chromatic polynomial $P_{H}(G, t) = P(G, t)$. 

Question: Are there easy proofs that $P_{H}(G, t)$ is a polynomial for $t \geq 0$ ?