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… Expand

The quartic (i.e., 4-regular) multigraphs are characterized with the property that every edge lies in a triangle, and it is shown that a simple quartic graph with every edge in a Triangle is either the square of a cycle, the line graph of a cubic graph or a graph obtained from the line multigraph of an cubic multigraph by replacing copies of K1, 1, 3.Expand