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Table 1 Zumkeller numbers used to label the edges of \(SSD(P_{m}\times P_{n})\)

From: k-Zumkeller labeling of super subdivision of some graphs

Edges Labels Num. of labels
\(u_{1j}u_{1,j(j+1)}^{k}\) \(p2^{(2mn-m-n-1)t+k}\) \((n-1)t\)
\(u_{ij}u_{i,j(j+1)}^{k}\),
\(u_{(i-1)j}u_{(i-1)i,j}^{k}\),
\(2\le i\le m\)
\(p2^{(2mn-m-1)t-(n-1)ti+k}\) \((m-1)(n-1)t\)
\(u_{i,j(j+1)}^{k}u_{i(j+1)}\),
\(u_{i(i+1),j}^{k}u_{(i+1)j}\),
\(1\le i\le m-1\)
\(p2^{(2mn-m-1)t-(n-1)ti+k}\) \((m-1)(n-1)t\)
\(u_{m,j(j+1)}^{k}u_{m(j+1)}\) \(p2^{(mn-1)t+k}\) \((n-1)t\)