Q:

N-dimensional Hypercubes coloring

How many ways 3-cube vertices can be coloring using 10 color, vertices which have relation is not able to have same color.
I would also appreciate anyone who show the solution for finding total ways problems in order to color the graphs with more color than chromatic number.

A:

Developing the answer of Casteels, this can be done with Maple. The Maple code $$with(GraphTheory): with(SpecialGraphs):G := PrismGraph(4): $$ $$ChromaticPolynomial(G, t); eval(\%, t = 10); $$ outputs $$ t \left( t-1 \right)  \left( {t}^{6}-11\,{t}^{5}+55\,{t}^{4}-159\,{t}^
{3}+282\,{t}^{2}-290\,t+133 \right)
 $$ and $28478970. $
See PrismGraph and ChromaticPolynomial for info.

