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

TL;DR

This paper proves the existence of $d$-divisible graceful $ ext{alpha}$-labelings for the graph $C_{4k} imes P_m$ across various parameters, extending the concept's applicability to cyclic graph decompositions.

Contribution

It establishes the existence of $d$-divisible graceful $ ext{alpha}$-labelings for $C_{4k} imes P_m$ for multiple values of $d$, generalizing previous concepts.

Findings

Existence of $d$-divisible graceful $ ext{alpha}$-labelings for $C_{4k} imes P_m$

Extension of labelings to new classes of graphs

Potential applications in cyclic graph decompositions

Abstract

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 $k\geq1$, $m\geq2$ for several values of $d$.