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 $f(V(G))=\{r,r+1,\dots ,r+p-1\}$ ($0\le r<r+p-1\le n-1$) and the induced edge labeling $f^{\ast }:E(G)\to \{0,1,\dots ,n-1\}$ given by $$f^{\ast }(u,v)=f(u)+f(v),\text{for every edge }(u,v)$$ forms a sequence of distinct consecutive integers $\{k,k+1,\dots ,n+k-1\}$ for some $k$ ($1\le k\le n-2$). By utilizing the graphs having $2$-sequential labeling, several new families of sequential graphs are presented.