5. Graph-Coloring. Mapmakers try to use as few colors as possible when coloring countries on a map, as long as no two countries that share a border have the same color. We can model this problem with an undirected graph G = (V,E) in which each vertex represents a country and vertices whose respective countries share a border are adjacent. A k-coloring is a function c: V -> {1, 2, , k} such that c(u) c(v) for every edge (u,v) E. In other words the number 1, 2, .., k represent the k colors and adjacent vertices must have different colors. The graph-coloring problem is to determine the minimum number of colors needed to color a given graph. a. State the graph-coloring problem as a decision problem K-COLOR. Show that your decision problem is solvable in polynomial time if and only of the graph-coloring problem is solvable in polynomial time. b. It has been proven that 3-COLOR is NP-complete by using a reduction from SAT. Use the fact that 3-COLOR is NP-complete to prove that 4-COLOR is NP-complete.

5. Graph-coloring. Mapmakers try to use as few colors as possible when coloring countries on a map, as long as no two countries that share a border have the same color. We can model this problem with an undirected graph G J (VLE) n which each vertex represents a country and vertices whose respective countries share a border are adjacent. A k-coloring is a function c V 11, 2, kh such that c(u) C (v) for every edge (uwv) E E. In other words the number 1, 2 k represent the k colors and adjacent vertices must have different colors. The graph-coloring problem is to determine the minimum number of colors needed to color a given graph. a. State the graph-coloring problem as a decision problem K-COLOR. Show that your decision problem is solvable in polynomial time if and only of the graph-coloring problem is solvable in polynomial time b. t has been proven that 3-COLOR is plete by using a reduction from SAT. Use the fact that 3-COLOR is NP-complete to prove that 4-COLOR is NP-complete