The Operator $\Psi$ for the Chromatic Number of a Graph

• Authors: Nebojša Gvozdenović;Monique Laurent
• Venue: SIAM Journal on Optimization
• Year: 2008

We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi(\overline G)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\overline G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if $\beta(G)$ is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number $\chi^*(\cdot)$ and $\chi(\cdot)$ unless $\text{P} = \text{NP}$. Moreover, based on the Motzkin-Straus formulation for $\alpha(G)$, we give (quadratically constrained) quadratic and copositive programming formulations for $\chi(G)$. Under some mild assumptions, $n/\beta(G)\le \Psi_\beta(G)$, but, while $n/\beta(G)$ remains below $\chi^*(G)$, $\Psi_\beta(G)$ can reach $\chi(G)$ (e.g., for $\beta(\cdot)=\alpha(\cdot)$). We also define new polynomial time computable lower bounds for $\chi(G)$, 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. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592-615].