Ascending Subgraph Decompositions of Oriented Graphs that Factor into Triangles

In 1987, Alavi, Boals, Chartrand, Erdős, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). In a previous paper, Wagner showed that all oriented complete balanced tripartite graphs have an ASD. In this paper, we will show that all orientations of an oriented graph that can be factored into triangles with a large portion of the triangles being transitive have an ASD. We will also use the result to obtain an ASD for any orientation of complete multipartite graphs with 3 partite classes each containing 2 vertices (a (2 : 3 )) or 4 vertices (a (4 : 3 )).