A new property of the Lovász number and duality relations between graph parameters

We show that for any graph G, by considering “activation” through the strong product with another graph H , the relation αpGq ď θpGq between the independence number and the Lovász number of G can be made arbitrarily tight: Precisely, the inequality αpG ˆHq ď θpGˆHq “ θpGqθpHq 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 αpGˆHq ď αpGqαpHq, with the fractional packing number α ̊pGq, 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 α and α are dual to each other, and the Lovász number θ is self-dual. We also show results towards the duality of Schrijver’s and Szegedy’s variants θ and θ of the Lovász number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.