Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares *,**

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

Automated summary

This paper formulates Vizing's conjecture as a mathematical question and uses methods such as semidefinite programming to obtain certificates, successfully confirming the conjecture for specific classes of graphs. This approach also applies to larger graph classes, providing possibilities for generalizing the conjecture.