Extremal number of arborescences

TL;DR

This paper investigates extremal problems for the number of arborescences in Eulerian orientations of graphs, providing exact solutions for specific graph classes and bounds for general graphs, with implications for graph orientation optimization.

Contribution

It offers the first exact minimization results for arborescences in Eulerian orientations of complete and bipartite graphs, and establishes bounds for arbitrary graphs.

Findings

Minimization of arborescences in complete graphs and bipartite graphs.

A stronger lower bound for arborescences in tournaments based on out-degree sequences.

An upper bound for arborescences in Eulerian orientations of arbitrary graphs.

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 $$\mathrm{allarb}(T)\geq \frac{1}{n}\left(\prod_{k=1}^n(d^+_k+1)+\prod_{k=1}^nd^+_k\right),$$ 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 $n$.