It is known that, for any positive integer n, the path \(P_n\) of order n satisfies the Vizing’s conjecture, that is, for any graph G, \(\gamma (G\square P_n)\ge \gamma (G)\gamma (P_n)\) where \(\gamma \) stands for the domination number and \(\square \) for the Cartesian product. In an attempt to prove Vizing’s conjecture, Clark and Suen proved in 2000 that \(\gamma (X\square Y)\ge \frac{1}{2}\gamma (X)\gamma (Y)\) for any pair of graphs X and Y. Combining these two inequalities, we have \(\gamma (X\square Y\square P_n)\ge \frac{1}{2}\gamma (X)\gamma (Y)\gamma (P_n).\) In this paper, we use space projections to build a general framework to obtain a lower bound for the domination number of a triple Cartesian product of graphs. As an application, we show that \(\gamma (X\square Y\square P_{n})\ge c_n\gamma (X)\gamma (Y)\gamma (P_{n}),\) where \(c_n\) is almost \(\frac{3}{4}\) when n is big enough.
