On $d$-divisible graceful $\alpha$-labelings of $C_{4k}\times P_m$

In a previous paper the concept of a $d$-divisible graceful $\alpha$-labeling has been introduced as a generalization of classical $\alpha$-labelings and it has been shown how it is useful to obtain certain cyclic graph decompositions. In the present paper it is proved the existence of $d$-divisible graceful $\alpha$-labelings of $C_{4k}\times P_m$ for any integers $k\ge 1$, $m\ge 2$ for several values of $d$.