Zumkeller labeling of some path related graphs

Abstract

A positive integer $n$ is said to be Zumkeller number if its positive factors can be partitioned into 2 disjoint parts with the equal sum, that is each part with sum $\sigma(n) / 2$. Let $G=(V(G), E(G))$ be a graph. An one to one function $f$ defined on $V(G)$ to a subset of natural numbers is termed as Zumkeller labeling of $G$ if the induced function $f^*: E(G) \rightarrow \mathbb{N}$ defined as $f^*(x y)=f(x) f(y)$ assigns a Zumkeller number for all $x y \in E(G), x, y \in V(G)$. A graph $G=(V(G), E(G))$ admits Zumkeller labeling is called a Zumkeller graph. In this manuscript, we investigate Zumkeller labeling for several classes of path graph.