The IsVertexColorable(G,k) function returns true if the graph G is k-colorable and false otherwise.  That is, if the vertices of G can be colored with k colors such that no two adjacent vertices have the same color.

If an optional argument d is specified, then IsVertexColorable(G,k,d) returns true if the graph G is (k,d) colorable, and false otherwise. That is, it returns true if the vertices of G can be colored with k colors such that two vertices with any given color are at least distance d apart.  When d is not specified it is assumed to be 1.

If a name col is specified, then this name is assigned the list of colors of a coloring of the vertices of G, if it exists.

The algorithm first tries a greedy coloring of the vertices of G starting with a maximum clique in G.  If this fails to find a k-coloring it does an exhaustive search using a backtracking algorithm.  The problem of testing if a graph is k-colorable is NP-complete, meaning that no efficient (polynomial time) algorithm is known.  The exhaustive search will take exponential time on some graphs.