The Operator Ψ for the Chromatic Number of a Graph ∗

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 Ψ β (G), nested between α(G) and χ(G); Ψ β (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. Moreover, based on the Motzkin–Straus formulation for α(G), we give (quadratically constrained) quadratic and copositive programming formulations for χ(G). Under some mild assumptions, n/β(G) ≤ Ψ β (G), but, while n/β(G) remains below χ * (G), Ψ β (G) can reach χ(G) (e.g., for β(·) = α(·)). We also define new polynomial time computable lower bounds for χ(G), improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. 1. Introduction. The chromatic number χ(G) of a graph G = (V, E) is the minimum number of colors needed to color the nodes of G in such a way that adjacent nodes receive distinct colors. Computing χ(G) is an NP-hard problem [11], and it is also hard to approximate χ(G) within |V (G)| 1/14− for any > 0 [1]. An obvious lower bound for χ(G) is the clique number ω(G), defined as the maximum size of a clique (i.e., a set of pairwise adjacent nodes) in G; computing ω(G) is also hard [11] as well as approximating ω(G) within |V (G)| 1/6− for any > 0 [1]. A well-known stronger lower bound for χ(G) is ϑ(G) := ϑ G , the theta number of the complementary graph, introduced by Lovász [23] (see (2.3)). The theta number satisfies the " sandwich inequality " :