  Considering regular graphs with every edge in a triangle we prove lower
bounds for the number of triangles in such graphs. For r-regular graphs with r
<= 5 we exhibit families of graphs with exactly that number of triangles and
then classify all such graphs using line graphs and even cycle decompositions.
Examples of ways to create such r-regular graphs with r >= 6 are also given. In
the 5-regular case, these minimal graphs are proven to be the only regular
graphs with every edge in a triangle which cannot have an edge removed and
still have every edge in a triangle.
