I can use various algorithms to list all proper $k$-colorings of the vertices of the ladder rung graph $nP_2$, the first six are show below.
Is there a quick way to list all proper $k$-colorings of the vertices?
The chromatic polynomial can count them easily (i.e. is quick to compute).
But does the nature of the graph present a coloring technique which improves the brute force method of checking every possible coloring for at least one pair of similar, adjacent colors, rejecting, and then keeping the rest?
So far, even for $n=6$, listing all $6$-colorings takes weeks of computational time using the best known graph coloring algorithms in Mathematica 12.