  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 these 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, the line graph of a cubic graph or a graph
obtained from the line multigraph of a cubic multigraph by replacing triangles
with copies of K_{1,1,3}.
