On edge irregularity strength of cycle-star graphs

Abstract

For a simple graph $G$, a vertex labeling $\varphi :V(G)\to \{1,2,\dots ,k\}$ is called $k$-labeling. The weight of an edge $uv$ in $G$, written $w_{\varphi }(uv)$, is the sum of the labels of end vertices $u$ and $v$, i.e., $w_{\varphi }(uv)=\varphi (u)+\varphi (v)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two distinct edges $u$ and $v$, $w_{\varphi }(u)\ne w_{\varphi }(v)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we study the edge irregular $k$-labeling for cycle-star graph $C{S}_{k,n-k}$ and determine the exact value for cycle-star graph for $3\le k\le 7$ and $n-k\ge 1$. Finally, we make a conjecture for the edge irregularity strength of $C{S}_{k,n-k}$ for $k\ge 8$ and $n-k\ge 1$.

14 pages, 10 figures