New Families of Sequential Graphs

Abstract

Let \(G = (V(G), E(G))\) be a finite simple graph with \(p\) vertices and \(n\) edges. A labeling of \(G\) is an injection \(f: V(G) \to {Z}_n\). A labeling of \(G\) is called \(2\)-sequential if \(f(V(G)) = \{r, r+1, \ldots, r+p-1\}\) (\(0 \leq r <r+ p-1 \leq n-1\)) and the induced edge labeling \(f^*: E(G) \to \{0, 1, \ldots, n-1\}\) given by \[f^*(u,v) = f(u) + f(v), \quad \text{for every edge } (u,v) \] forms a sequence of distinct consecutive integers \(\{k, k+1, \ldots, n+k-1\}\) for some \(k\) (\(1 \leq k \leq n-2\)). By utilizing the graphs having \(2\)-sequential labeling, several new families of sequential graphs are presented.