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

TL;DR

This paper investigates the duality relations between various graph parameters, demonstrating how the Lovász number can be made arbitrarily tight with the independence number through graph products, revealing new dualities among these parameters.

Contribution

It introduces new duality relations between graph parameters like the Lovász number, independence number, and fractional packing number, using graph products to establish asymptotic equalities.

Findings

The relation lpha(G)   (G) can be made arbitrarily tight via strong product with suitable graphs.

The independence number and fractional packing number are dual under certain graph products.

Duality relations are extended to variants of the Love1sz number and chromatic number under different graph products.

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.