Table 2 Zumkeller numbers used to label the edges of \(SSD(C_{m}\times P_{n})\)
From: k-Zumkeller labeling of super subdivision of some graphs
Edges | Num. of edges | Labels | Num. of labels |
|---|---|---|---|
\(u_{ij}u_{i,j(j+1)}^{k}\), \(u_{in}u_{i,n1}^{k}\) \(1\le i\le m\), \(1\le j\le n-1\) | mnt | \(p2^{(2mn-1)t-nti+k}\) | mt |
\(u_{i,j(j+1)}^{k}u_{i(j+1)}\) \(1\le i\le m\), \(1\le j\le n-1\) \(u_{(i-1)j}u_{(i-1)i,j}^{k}\) \(2\le i\le m\), \(2\le j\le n-1\) \(u_{(i-1),n1}^{k}u_{(i-1)1}\), \(2\le i\le m\) \(u_{i(i+1),j}^{k}u_{(i+1)j}\) \(1\le i\le m\), \(2\le j\le n-1\) \(u_{(i-2)1}u_{(i-2)(i-1),1}^{k}\) \(3\le i\le m\), \(1\le j\le n-1\) \(u_{(i-1)i,1}^{k}u_{i1}\), \(2\le i\le m\) | \((n-1)mt\) \((n-1)(m-1)t\) \((m-1)t\) \((m-1)t\) \((m-2)t\) \((m-1)t\) | \(p2^{2mnt-nti+k }\) | mt |
\(u_{m,n1}^{k}u_{m1}\), \(u_{(m-1)1}u_{(m-1)m,1}^{k}\) | 2t | \(p2^{mnt-nt+k}\) | t |