Approximating the Chromatic Number of a Graph by Semidefinite Programming∗

We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ G , to a new graph parameter Ψβ(G), nested between ω(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number χ∗(·) and χ(·) unless P=NP and, based on Motzkin-Straus formulation for α(G), we give quadratic and copositive programming formulations for χ(G). Under some mild assumption, n/β(G) ≤ Ψβ(G) but, while n/β(G) remains below χ∗(G), Ψβ(G) can reach χ(G) (e.g., for β(·) = α(·)). We define new lower bounds for χ(G) which we test on Hamming graphs and on some benchmark graphs. Our preliminary experimental results indicate that the new bounds can be much stronger than the classic bound θ G (and its strengthenings obtained by adding nonnegativity and triangle inequalities).