An Improved Bound for Upper Domination of Cartesian Products of Graphs

In this paper, we prove a problem proposed by Bre\v{s}ar: for any graphs $G$ and $H$, $\Gamma(G\square H)\ge\Gamma(G)\Gamma(H)+ \min{|V(G)|-\Gamma(G),|V(H)|-\Gamma(H)}$,where$\Gamma(G)$ denotes the upper domination number of $G$.