The chromatic number of a graph G is the minimum k such that it has a k-coloring. As we saw in Chapter, it is NP-complete for k ≥ 3 to decide whether a given input graph has chromatic number ≤ k.
(a) Show that for every natural number w ≥ 1, there is a number k (w) so that the following holds. If G is a graph of tree-width at most w, then G has chromatic number at most k (w). (The point is that k (w) depends only on w, not on the number of nodes in G.)
(b) Given an undirected n-node graph G = (V, E) of tree-width at most w, show how to compute the chromatic number of G in time O (f (w) • p (n)), where p (n) is a polynomial but f (•) can be an arbitrary function.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.