On the Cordiality of the \(t\)-Uniform Homeomorphs – \(II\) (Complete Graphs)

Abstract

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(\overline{f}: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(f(uv) = |f(u) – f(v)|\). 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 \(|v_f(0) – vf(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).

A \(t\)-uniform homeomorph \(P_t(G)\) of \(G\) is the graph obtained by replacing all edges of \(G\) by vertex disjoint paths of length \(t\). In this paper we show that (1)\(P_t(K_{2n})\) is cordial for all \(t \geq 2\).(2) \(P_t(K_{2n+1})\) is cordial if and only iff (a) \(t \equiv 0 \pmod{4}\), or(b) \(t\) is odd and \(n\) is not \(\equiv 2 \pmod{4}\), or (c) \(t \equiv 2 \pmod{4}\) and \(n\) is even.