# Non-equivalent eulerian trails in $K_{2n+1}$

Two eulerian trails of $$K_{2n+1}$$ are defined to be equivalent if the orientations obtained by orienting the edges as traversed by the trails are isomorphic as digraph. How many non-equivalent trails are there?

• The eulerian trails $0\to 1\to 2\to 0$ and $0\to 2\to 1\to 0$ of the complete graph $K_3$ are considered non-equivalent? – Freddy Barrera Apr 22 at 12:28
• @FreddyBarrera Those would be considered equivalent since they are isomorphic as digraphs (both are just directed triangles). – Tony Huynh Apr 22 at 13:14
• @FreddyBarrera, yes they are the same. Number for $K_3$ is 1. – hbm Apr 22 at 13:49

The asymptotic number is known. It is $$RT(2n+1)/(2n+1)!$$, where $$RT(2n+1)$$ is given in the abstract of this paper.