  In recent papers by Grohe and Marx, the treewidth of the line graph of the
complete graph is a critical example. We determine the exact treewidth of the
line graph of the complete graph. By extending these techniques, we determine
the exact treewidth of the line graph of a regular complete multipartite graph.
For an arbitrary complete multipartite graph, we determine the treewidth of the
line graph up to a lower order term.
