What's the maximum number of distinct triangles in complete graph of n vertices such that each edge of the graph appears in exactly two triangles? Maybe with the way to construct them?
I hope I phrase the question correctly in the title. For example, for $K_6$, I obtain 10 triangles:
123, 124, 236, 345, 346, 245, 135, 156, 256, 164
which I do not think more can be added. I tried to use various decompositions but I still got nothing. Does anyone know any link or hint that might help? Thanks