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 $|v_f(0)–vf(1)|\le 1$ and $|e_f(0)–e_f(1)|\le 1$.

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 $t\ge 2$.(2) $P_t(K_{2n+1})$ is cordial if and only iff (a) $t\equiv 0(\mathrm{mod}4)$, or(b) $t$ is odd and $n$ is not $\equiv 2(\mathrm{mod}4)$, or (c) $t\equiv 2(\mathrm{mod}4)$ and $n$ is even.