Balancing connected colourings of graphs

TL;DR

This paper proves that graphs with two edge-disjoint spanning trees can be coloured with two colours to ensure connectedness and balanced degrees, improving previous results and exploring related variations.

Contribution

It introduces a new colouring method for such graphs, achieving tighter degree balance and addressing open questions from prior research.

Findings

Edges of graphs with two edge-disjoint spanning trees can be coloured to keep both colour subgraphs connected.

The degree difference at each vertex can be limited to at most four.

The paper resolves two open questions negatively, expanding understanding of graph colourings.

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 H\"orsch. We discuss variations of the question for digraphs, infinite graphs and a computational question, and resolve two further questions of H\"orsch in the negative.