Properly colored graph (edge has color) means that any two adjacent edges have distinct colors.
For any graph with $m$ edges such that it can be properly colored using $k$ colors. What is the minimum number of $k$-colorings of a graph with $m$ edges, over all graphs that admit such a coloring?
For example, If we construct the graph as a matching with $m$ edges, then the number of possible properly colored matchings is $k^{m}$. Thus, $k^{m}$ is an upper bound.
For the above problem, we have the following result:
Thm. For any m, if $k \ge \frac{6}{7}m$, then the minimum number of $k$-colorings of a graph with $m$ edges is at least $\frac{k^m}{e^{m+2}}$.
Now we consider the following problem:
We define $H^k(m)$ as the number of $k$-colorings of a graph $H$ with $m$ edges(indeed, this is edge chromatic-polynomial). For an integer $s\le m$, if we fix $s$ edges's colors in $H$, then define $H^k_s(m)$ as the number of $k$-colorings of the subgraph of $H$ with remain edges such that $H$ is properly colored. Then do we have $$ \frac{H^k(m)}{H^k_s(m)} \ge (ck)^s? $$ for some constant c.
Remark: we only consider the case that $H^k_s(m) > 0$ and $H$ has no multi-edge. Of course, we may need $k \ge \frac{6}{7}m$ or others.
Or are there any research results on $H^k_s(m)$?
