On the Edge-graceful Spectra of the Cylinder Graphs (I)

Let \( G \) be a \( (p, q) \)-graph and \( k \geq 0 \). A graph \( G \) is said to be k-edge-graceful if the edges can be labeled by \( k, k+1, \dots, k+q-1 \) so that the vertex sums are distinct, modulo \( p \). We denote the set of all \( k \) such that \( G \) is \( k \)-edge graceful by \( \text{egS}(G) \). The set is called the \textbf{edge-graceful spectrum} of \( G \). In this paper, we are concerned with the problem of exhibiting sets of natural numbers which are the edge-graceful spectra of the cylinder \( C_{n} \times P_{m} \), for certain values of \( n \) and \( m \).