On \(d\)-divisible \(\alpha\)-labelings of \(C_{4k} \times P_m\) (Q2848794)

Let \(\Gamma = (V,E)\) be a graph of size \(e = d \cdot m\). A \(d\)-divisible graceful labeling of \(\Gamma\) is an injection \(f : V \rightarrow \{0,1,2,d(m+1)-1\}\) such that NEWLINE\[NEWLINE\{|f(x)-f(y)|\text{ }|xy\in E\}=\{1,2,3,\dots,d(m+1)\}\setminus \{k(m+1)|k=1,2,\dots,d\}.NEWLINE\]NEWLINENEWLINEA \(d\)-divisible \(\alpha\)-labeling of a bipartite graph \(\Gamma=(V_1,V_2,E)\) is a \(d\)-divisible graceful labeling of \(\Gamma\) such that \(\max\{f(x)|x\in V_1\} < \min\{f(x)|x\in V_2\}\). In the paper, the existence of \(d\)-divisible \(\alpha\)-labelings of \(C_{4k} \times P_m\) is determined. The results are proved by induction in \(m\).