A new property of the Lovász 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 α(G)≤ϑ(G) between the independence number and the Lovász number of G can be made arbitrarily tight: Precisely, the inequality α(G⊠H)⩽ϑ(G⊠H)=ϑ(G)ϑ(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 α(G⊠H)⩽α∗(G)α(H), with the fractional packing number α∗(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 α and α∗ are dual to each other, and the Lovász number ϑ is self-dual. We also show 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.