On the Cordiality of the \(t\)-ply \(P_t(u, v)\).

Abstract

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(f: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) 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 give necessary and sufficient conditions for the cordiality of the \(t\)-ply \(P_t(u,v)\), i.e. a thread of ply number \(t\).