A general partition graph (gpg) is an intersection graph G on a set S so that for every maximal independent set M of vertices in G, the subsets assigned to the vertices in M partition S. These graphs have been characterized by the presence of special clique covers. The Triangle Condition T for a graph G is that for any maximal independent set M and any edge uv in G M, there is a vcrtex W E M so that uvw is a triangle in G. Condition T is necessary but not sufficient for a graph to be a gpg and a computer search has found the smallest ten counterexamples, one with nine vertices and nine with ten verticcs. Any non-gpg satisfying Condition T is shown to induce a required subgraph on six vertices, and a method of generating an infinite class of such graphs is described. The main result establishes the equivalence of the following conditions in a chordal graph G: (i) G is a gpg (ii) G satisfies Condition T (iii) every edge in G is in an end-clique. The result is extended to a larger class of graphs.