Globally balancing spanning trees

TL;DR

This paper proves that in graphs with multiple edge-disjoint spanning trees, it is possible to select such trees with balanced degrees at each vertex, confirming a conjecture and providing bounds related to the number of trees.

Contribution

The authors establish the existence of balanced edge-disjoint spanning trees in graphs, confirming Kriesell's conjecture and extending results to any number of trees with logarithmic degree bounds.

Findings

For graphs with two edge-disjoint spanning trees, balanced trees with degree difference at most 5 are possible.

For any number k of edge-disjoint spanning trees, degree differences are bounded by a constant c_k in O(log k).

The results resolve a longstanding conjecture in graph theory.

Abstract

We show that for every graph $G$ that contains two edge-disjoint spanning trees, we can choose two edge-disjoint spanning trees $T_1,T_2$ of $G$ such that $|d_{T_1}(v)-d_{T_2}(v)|\leq 5$ for all $v \in V(G)$. We also prove the more general statement that for every positive integer $k$, there is a constant $c_k \in O(\log k)$ such that for every graph $G$ that contains $k$ edge-disjoint spanning trees, we can choose $k$ edge-disjoint spanning trees $T_1,\ldots,T_k$ of $G$ satisfying $|d_{T_i}(v)-d_{T_j}(v)|\leq c_k$ for all $v \in V(G)$ and $i,j \in \{1,\ldots,k\}$. This resolves a conjecture of Kriesell.