Quartic Graphs with Every Edge in a Triangle

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from the line multigraphs of cubic multigraphs by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every edge in a triangle is either the square of a cycle or a graph obtained from the line graph of a cubic graph by replacing triangles with copies of K1,1,3.