Digraphs with exactly one Eulerian tour

TL;DR

This paper provides combinatorial proofs for counting loopless digraphs with exactly one Eulerian tour, establishing a connection with Catalan numbers, rooted plane trees, and parenthesis arrangements.

Contribution

It introduces two combinatorial proofs and bijections that count such digraphs, linking them to well-known combinatorial structures.

Findings

Number of such digraphs is (1/2)(n-1)!C_n.

Established bijections with rooted plane trees and parenthesis arrangements.

Provided combinatorial proofs for the enumeration.

Abstract

We give two combinatorial proofs of the fact that the number of loopless digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes the $n$-th Catalan number. We construct a bijection with a set of labeled rooted plane trees and with a set of valid parenthesis arrangements.