Randomized $\Delta$-Edge-Coloring via Quaternion of Complex Colors

This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing’s theorem states that the edge coloring of a simple graph G requires either ∆ or ∆ + 1 colors, where ∆ is the maximum vertex degree of G. Holyer proved that it is NP-complete to decide whether G is ∆-edge-colorable even for cubic graphs. By introducing the concept of complex colors, we show that the color-exchange operation follows the same multiplication rules as quaternion. An initially ∆-edge-colored graph G allows variable-colored edges, which can be eliminated by color 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 ∆-edge-colored graph is reached. For a randomly generated graph G, we prove that our algorithm returns a proper ∆-edge-coloring with a probability of at least 1/2 in O(∆|V ||E|) time if G is ∆-edge-colorable. Otherwise, the algorithm halts in polynomial time and signals the impossibility of a solution, meaning that the chromatic index of G probably equals ∆ + 1. Animations of the edge-coloring algorithms proposed in this paper are posted at YouTube http://www.youtube.com/watch?v=KMnj4UMYl7k.