Balancing connected colourings of graphs (Q2699645)

Summary: 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 \textit{F. Hörsch} [Eur. J. Comb. 109, Article ID 103644, 15 p. (2023; Zbl 1505.05039)]. We discuss variations of the question for digraphs, infinite graphs and a computational question, and resolve two further questions of Hörsch in the negative.