Triangle-free Graphs

Question

Is it possible to have triangles in the line graphs of triangle free graphs? I drew some examples for n=1,2,3,4, and 5 but I don't see any triangles yet.

And what is the explanation behind it if there are or aren't triangles?

Answer 1

Sure -- if the original graph has a vertex of degree 3, then there will be a triangle in the line graph.

The smallest example would be

      O
     /
O---O
     \
      O

whose line graph is a triangle.

Or, for a bridgeless example, consider the cube graph.