Question: 4. (6 points) Coloring a graph The minimum coloring problem asks to label the vertices with the fewest number of distinct colors so that no

4. (6 points) Coloring a graph The minimum coloring problem asks to label the vertices with the fewest number of distinct colors so that no edge has two endpoints having the same color. (a) (2 points) Consider the following greedy algorithm: find a maximum independent set, color it with color 1, remove it from the graph, and repeat (using colors 2 2). Show that this algorithm is suboptimal by providing a graph of size at most 5 where it does not produce the optimal coloring. (b) (3 points) Give an O(4") time algorithm for finding the fewest number of colors needed to color a graph. One way that this can be done is to define a state to be the minimum number of colors needed to color a subset of the vertices, and the transition can then be an entire subset of vertices that can be colored using the same color. (c) (1 points): Improve the running time to 0(3")

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