5.1.23. (!) Place n points on a circle, where n > k(k+1). Let Gn.k be the 2k-regular graph obtained by joining each point to the k nearest points in each direction on the circle. For example, Gx,1 = Cr, and G7,2 appears below. Prove that x(mk) = k +1 ifk + 1 divides n and x(n) = k +2 if k + 1 does not divide n. Prove that the lower bound on n cannot be weakened, by proving that x (Gxx+1)-1.4) >k +2 if k > 2. 5.1.23. (!) Place n points on a circle, where n > k(k+1). Let Gn.k be the 2k-regular graph obtained by joining each point to the k nearest points in each direction on the circle. For example, Gx,1 = Cr, and G7,2 appears below. Prove that x(mk) = k +1 ifk + 1 divides n and x(n) = k +2 if k + 1 does not divide n. Prove that the lower bound on n cannot be weakened, by proving that x (Gxx+1)-1.4) >k +2 if k > 2