Fact: Suppose that d is the maximum degree among all vertices in a graph G. Then G has a proper (d + 1)-coloring.

(a) Let G be a graph with O(|V |) edges. Recall that (G) is the chromatic number of G. Show that (G) = O( p |V |).

(b) Does there exist a graph with O(|V |) edges such that (G) = (p |V |)?