Sum-of-squares certificates for Vizing's conjecture via determining Gröbner bases

Abstract

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with nG vertices and domination number kG and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of kG, nG, kH, and nH.In this paper, we consider their approach for kG=kH=1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (nG+nH−1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with nG+nH−1=d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with kG=kH=1 and nG+nH≤15.