Immersing complete digraphs

We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever the minimum degree is at least t ( t − 1 ) , and, for t ≤ 4 , minimum degree at least t − 1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high indegree and outdegree which do not contain an immersion of the complete digraph on three vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construction of such graphs by Thomassen.