Cordial Labelings Of Some Wheel Related Graphs

Abstract

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\). A vertex labelling \(f : V \rightarrow \{0,1\}\) induces an edge labelling \(\overline{f} : E \rightarrow \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\). In this paper, we show that for every positive integer \(t\) and \(n\) the following families are cordial: (1) Helms \(H_{n}\). (2) Flower graphs \(FL_{n}\). (3) Gear graphs \(G_{n}\). (4) Sunflower graphs \(SFL_{n}\). (5) Closed helms \(CH_{n}\). (6) Generalised closed helms \(CH(t,n)\). (7) Generalised webs \(W(t, n)\).