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

In a previous paper the concept of a $d$-divisible graceful $α$-labeling has been introduced as a generalization of classical $α$-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 $α$-labelings of $C_{4k}\times P_m$ for any integers $k\geq1$, $m\geq2$ for several values of $d$.

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