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 domination numbers two graphs to number their Cartesian graph. In this paper, we formulate as a Positivstellensatz existence question. particular, select classes according vertices and encode an ideal/polynomial pair such that polynomial is non-negative on variety associated with ideal if only true for graph class. Using semidefinite programming obtain numeric sum-of-squares certificates, which then manage transform into symbolic certificates confirming non-negativity our polynomials. Specifically, exact low-degree sparse particular graphs. The obtained allow generalizations larger classes. Besides computational verification these more general also present theoretical proofs well conjectures questions further investigations.