Super edge-graceful paths

Abstract

A graph $G(V,E)$ of order $|V|=p$ and size $|E|=q$ is called super edge-graceful if there is a bijection $f$ from $E$ to $\{0,\pm 1,\pm 2,...,\pm \frac{q-1}{2}\}$ when $q$ is odd and from $E$ to $\{\pm 1,\pm 2,...,\pm \frac{q}{2}\}$ when $q$ is even such that the induced vertex labeling $f^*$ defined by $f^*(x) = \sum_{xy\in E(G)}f(xy)$ over all edges $xy$ is a bijection from $V$ to $\{0,\pm 1,\pm 2...,\pm \frac{p-1}{2}\}$ when $p$ is odd and from $V$ to $\{\pm 1,\pm 2,...,\pm \frac{p}{2}\}$ when $p$ is even. \indent We prove that all paths $P_n$ except $P_2$ and $P_4$ are super edge-graceful.