Prime cordial labeling of some wheel related graphs

Abstract

A prime cordial labeling of a graph $G$ with the vertex set $V(G)$ is a bijection $f: V(G) \rightarrow\{1,2,3, \ldots,|V(G)|\}$ such that each edge $u v$ is assigned the label 1 if $\operatorname{gcd}(f(u), f(v))=1$ and 0 if $\operatorname{gcd}(f(u), f(v))>1$, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1 . A graph which admits prime cordial labeling is called prime cordial graph. In this paper we prove that the gear graph $G_n$ admits prime cordial labeling for $n \geq 4$. We also show that the helm $H_n$ for every $n$, the closed helm $C H_n$ (for $n \geq 5$ ) and the flower graph $F l_n$ (for $n \geq 4$ ) are prime cordial graphs.