Koh and Tay proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{ C}_0, \mathscr{ C}_1$ and $\mathscr{ C}_2$. In this paper, we prove that vertex-multiplications of cartesian products of graphs $G\times H$ lie in $\mathscr{ C}_0$ ($\mathscr{ C}_0\cup \mathscr{ C}_1$ resp.) if $G^{ (2)}\in \mathscr{ C}_0$ ($\mathscr{ C}_1$ resp.), $d(G)\ge 2$ and $d(G\times H)\ge 4$. We also focus on cartesian products involving trees, paths and cycles and show that most of them lie in $\mathscr{ C}_0$.
Categories: math.CO