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 $\chi\left( {\ol G} \right)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\ol G)$ 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 P=NP. Moreover, based on Motzkin-Straus formulation for $\alpha(G)$, we give (quadratically constrained) quadratic and copositive programming formulations for $\chi(G)$. Under some mild assumption, $n/\beta(G)\le \Psi_\beta(G)$ 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 Lov\'{a}sz 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.

Additional Metadata
Keywords (fractional) chromatic number, stability number, Lov\' asz theta number, semidefinite programming
MSC Semidefinite programming (msc 90C22), Combinatorial optimization (msc 90C27)
THEME Logistics (theme 3)
Publisher S.I.A.M.
Journal SIAM Journal on Optimization
Project Semidefinite programming and combinatorial optimization
Gvozdenovic, N, & Laurent, M. (2008). The operator $\Psi$ for the Chromatic Number of a Graph. SIAM Journal on Optimization, 19(02), 572–591.