Towards a Computational Proof of Vizing's Conjecture using Semidefinite Programming and Sums-of-Squares

Abstract

Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs.The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations.An extended abstract containing the ideas of this optimization-based approach for tackling Vizing's conjecture appeared as [10]. This article now also contains the full and complete proofs, new certificates only conjectured in the extended abstract, and further theoretical and computational results for particular cases. This also lead to a restructuring of the whole article, and more examples and many more remarks explaining the implications of the results are provided. The authors gratefully acknowledge the support of Fulbright Austria (via a Visiting Professorship at