Globally balancing spanning trees

We show that for every graph G that contains two edge-disjoint spanning trees, we can choose two edge-disjoint spanning trees T1, T2 of G such that |dT1(v) - dT2(v)| <= 5 for all v E V(G). We also prove the more general statement that for every positive integer k, there is a constant ck E O(log k) such that for every graph G that contains k edge-disjoint spanning trees, we can choose k edge-disjoint spanning trees T1, ... , Tk of G satisfying |dTi(v)- dTj(v)| <= ck for all v E V(G) and i,j E {1,. ..,k}. This resolves a conjecture of Kriesell. (c) 2022 Elsevier Ltd. All rights reserved.