Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and let m be the minimum degree of the vertices of G. Show that 2e/v≤ M. Step 1 The degree of each vertex is ≤ M, and there are v vertices. The degree of each vertex is ≤ M, and there are v vertices. Step 2 We know that 2e is the sum of the degrees of the vertices. Step 3 The sum of the degrees of the vertices must be ≤ to vM. Hence, 20s vM. Therefore, 2e/vs M. The sum of the degrees of the vertices must be ≤ to vM. We know that 2e is the sum of the degrees of the vertices. Step 4 Hence, 2e s vM. Therefore, 2e/vs M. The sum of the degrees of the vertices must be ≥ VM. Suppose that there are five young women and six young men on an island. Each woman is willing to marry some of the men on the island and each man is willing to marry any woman who is willing to marry him. Suppose that Anna is willing to marry Jason, Larry, and Matt; Barbara is willing to marry Kevin and Larry; Carol is willing to marry Jason, Nick, and Oscar; Diane is willing to marry Jason, Larry, Nick, and Oscar; and Elizabeth is willing to marry Jason and Matt. Identify the correct statements with respect to the matching between men and women formed in the previous question. (Check all that apply.) Check All That Apply It is a complete matching from the set of women to the set of men. It is not a complete matching from the set of women to the set of men. It is a maximum matching, since complete matching is always a maximum matching. × × It is not a maximum matching, because complete matching need not be always a maximum matching. ×