A Grover search-based algorithm for the list coloring problem

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ $k\ge 5$. The list problem asks following more general question: given available colors each vertex in graph, does it admit proper coloring? We propose quantum based Grover search to quadratically speed up exhaustive search. Our loses complexity ones specific restricted cases, but improves cases where lists considered are arbitrary nature.