Randomized Δ-edge colouring via exchanges of complex colours

Abstract

This paper explores the application of a new algebraic method of colour exchanges to the edge colouring of simple graphs. Vizing's theorem states that the edge colouring of a simple graph G requires either Δ or Δ+1 colours, where Δ is the maximum vertex degree of G. Holyer proved that it is NP-complete to decide whether G is Δ-edge colourable even for cubic graphs. By introducing the concept of complex coloured edges, we show that the colour-exchange operation of complex colours follows the same multiplication rules as quaternion. An initially Δ-edge-coloured graph G allows variable-coloured edges, which can be eliminated by colour exchanges in a manner similar to variable eliminations in solving systems of linear equations. The problem is solved if all variables are eliminated and a properly Δ-edge-coloured graph is reached. For Δ-regular uniform random graphs, we prove that our algorithm returns a proper Δ-edge colouring with a probability of 1−1/n in time if G is Δ-edge colourable. Otherwise, the algorithm halts in polynomial time and signals the impossibility of a solution, meaning that the chromatic index of G probably equals Δ+1.