Balancing connected colourings of graphs

Abstract

We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so that the blue and red graphs are connected and the blue and red degrees at each vertex differ by at most four. This improves a result of Hörsch. We discuss variations of the question for digraphs, infinite graphs and a computational question, and resolve two further questions of Hörsch in the negative.