Extremal number of arborescences

Abstract

In this paper we study the following extremal graph theoretic problem: Given an undirected Eulerian graph $G$, which Eulerian orientation minimizes or maximizes the number of arborescences? We solve the minimization for the complete graph $K_n$, the complete bipartite graph $K_{n,m}$, and for the so-called double graphs, where there are even number of edges between any pair of vertices. In fact, for $K_n$ we prove the following stronger statement. If $T$ is a tournament on $n$ vertices with out-degree sequence $d_1^{+},\dots ,d_n^{+}$, then $$\mathrm{allarb}(T)\ge \frac{1}{n}\left(\prod _{k=1}^n(d_k^{+}+1)+\prod _{k=1}^n{d}_k^{+}\right),$$ where $\mathrm{allarb}(T)$ is the total number of arborescences. Equality holds if and only if $T$ is a locally transitive tournament. We also give an upper bound for the number of arborescences of an Eulerian orientation for an arbitrary graph $G$. This upper bound can be achieved on $K_n$ for infinitely many $n$.