Globally balancing spanning trees

Balancing Connected Colourings of Graphs

“…We have shown that every double tree has a partition into two trees such that the degrees at each vertex differ by at most four (improving on Hörsch's [5] bound of five). Can this be further improved?…”

“…By repeatedly applying Theorem 1, Hörsch [5] obtained the bound c k " 22 log k. We could similarly derive improved bounds on c k and d k by repeatedly applying Theorem 3. When the requirement that each graph in the decomposition is a tree is dropped (so any k-edge colouring of the original graph G is allowed), a uniform bound on the colourdegree differences is attainable.…”

“…Consequently, upper bounds on c k are bounds on d k with a constant error term. It would be particularly interesting to resolve Hörsch's conjecture [5] of whether there is a uniform upper bound on the c k .…”

“…Kriesell [6] was the first to consider balancing colour-degrees in a blue/red-edge colouring of a double tree. Building on his work, Hörsch [5] gave the first constant bound when G is a double tree.…”