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

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Journal Article
Discrete Applied Mathematics, 2017, 216 pp. 489 - 501
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© 2016 Elsevier B.V. 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.
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