  Two Eulerian circuits, both starting and ending at the same vertex, are
avoiding if at every other point of the circuits they are at least distance 2
apart. An Eulerian graph which admits two such avoiding circuits starting from
any vertex is said to be doubly Eulerian. The motivation for this definition is
that the extremal Eulerian graphs, i.e. the complete graphs on an odd number of
vertices and the cycles, are not doubly Eulerian. We prove results about doubly
Eulerian graphs and identify those that are the `densest' and `sparsest' in
terms of the number of edges.
