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

Abstract

Let $G$ be a $(p,q)$-graph and $k\ge 0$. 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 $m$.