A Game Theoretic Approach for Efficient Graph Coloring

The game Γ(G) has always pure Nash equilibria. Each pure equilibrium is a proper coloring of G. Furthermore, there exists a pure equilibrium that corresponds to an optimum coloring.

We give a polynomial time algorithm \(\mathcal{A}\) which computes a pure Nash equilibrium of Γ(G).

The total number, k, of colors used in any pure Nash equilibrium (and thus achieved by \(\mathcal{A}\)) is \(k\leq\min\{\Delta_2+1, \frac{n+\omega}{2}, \frac{1+\sqrt{1+8m}}{2}, n-\alpha+1\}\), where ω, α are the clique number and the independence number of G and Δ ₂ is the maximum degree that a vertex can have subject to the condition that it is adjacent to at least one vertex of equal or greater degree. (Δ ₂ is no more than the maximum degree Δ of G.)

Thus, in fact, we propose here a new, efficient coloring method that achieves a number of colors satisfying (together) the known general upper bounds on the chromatic number χ. Our method is also an alternative general way of proving, constructively, all these bounds.

Finally, we show how to strengthen our method (staying in polynomial time) so that it avoids “bad” pure Nash equilibria (i.e. those admitting a number of colors k far away from χ). In particular, we show that our enhanced method colors optimally dense random q-partite graphs (of fixed q) with high probability.