A new property of the Lov\'asz number and duality relations between graph parameters

Abstract

We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $\alpha(G) \leq \vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$ can be made arbitrarily tight: Precisely, the inequality $$\alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H)$$ becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that $$\alpha(G \times H) \leq \alpha^*(G)\,\alpha(H),$$ with the fractional packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $\alpha$ and $\alpha^*$ are dual to each other, and the Lov\'{a}sz number $\vartheta$ is self-dual. We also show duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$ of the Lov\'{a}sz number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special issue in memory of Levon Khachatrian; v2 has a full proof of the duality between theta+ and theta- and a new author, some new references, and we corrected several small errors and typo