(5) Let G be a graph without loops. Define a cycle-free orientation of G to be an assignment of directions on each edge of G so that it has no directed cycles. For example, here is a cycle-free orientation of the triangle: Let (G) be the number of cycle-free orientations of G. For the triangle, we have 0(G) = 6 (there are 23-8 orientations total and 2 of them are directed cycles). If G has no edges, then we make the convention that (C)-1. (a) Let H be a graph without loops, and let H' be the simple version of H: H' has the same vertices of H, but if there is at least one edge between two vertices a and y in H, we replace all of them with a single edge in H'. Show that (b) Suppose G is simple. If e is an edge of G, show that (5) Let G be a graph without loops. Define a cycle-free orientation of G to be an assignment of directions on each edge of G so that it has no directed cycles. For example, here is a cycle-free orientation of the triangle: Let (G) be the number of cycle-free orientations of G. For the triangle, we have 0(G) = 6 (there are 23-8 orientations total and 2 of them are directed cycles). If G has no edges, then we make the convention that (C)-1. (a) Let H be a graph without loops, and let H' be the simple version of H: H' has the same vertices of H, but if there is at least one edge between two vertices a and y in H, we replace all of them with a single edge in H'. Show that (b) Suppose G is simple. If e is an edge of G, show that