Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case. As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated. The derived inequalities are tested on W the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems.