In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described. ©2004 American Mathematical Society.3571040914129Bertolim, M.A., Mello, M.P., De Rezende, K.A., Lyapunov graph continuation (2003) Ergod. Th. & Dynam. Sys., 23, pp. 1-58. , MR2004b:37042Bertolim, M.A., Mello, M.P., De Rezende, K.A., Poincaré-Hopf and Morse inequalities for Lyapunov graphs Ergod. Th. & Dynam. Sys., , To appear inConley, C., Isolated invariant sets and the Morse index (1978) CBMS Regional Conference Series in Mathematics, 38. , American Mathematical Society, Providence, RI, MR80c:58009Cruz, R.N., De Rezende, K.A., Gradient-like flows on high-dimensional manifolds (1999) Ergod. Th. & Dynam. Sys., 19 (2), pp. 339-362. , MR2001a:37026Fulkerson, D.R., Gross, O.A., Incidence matrices and interval graphs (1965) Pacific Journal of Mathematics, 15, pp. 835-855. , MR32:3881Hoffman, A.J., Kruskal, J.B., Integral boundary points of convex polyhedra (1956) Annals of Mathematics Studies, (38), pp. 223-246. , Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.), Princeton University Press, Princeton, NJ. MR18:980bMorse, M., Relations between the critical points of a real functions of n independent variables (1925) Trans. Am. Math. Soc., 27, pp. 345-396