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.