The chromatic polynomial $P(G, \lambda)$ gives the number of proper vertex colorings of the graph $G$ with $\lambda$ colors. I'm interested in how many possible colorings you loose when you add an edge $e$. The new graph might not be $\lambda$-colorable, but if it is still $\lambda$-colorable, is there any lower bound on $$\frac{P(G+e,\lambda)}{P(G,\lambda)}$$ or alternatively is there any upper bound on $\frac{P(G/e,\lambda)}{P(G,\lambda)}$, where $G/e$ denotes the edge contraction?
I found a similar result for deleting a vertex here, although they use the additional assumption that the number of colors is greater or equal to the number of vertices. Nevertheless this makes me optimistic that an interesting bound on the ratio above might be known?
