On \(Q(a)P(b)\)-Super Edge-Gracefulness of Hypercubes

Abstract

Let \( a, b \) be two positive integers. A \( (p, q) \)-graph \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful, or simply \( (a, b) \)-SEG, if there exist onto mappings \( f : E(G) \to Q(a) \) and \( f^* : V(G) \to P(b) \), where

\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even}, \\
\{0, \pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd},
\end{cases}
\]

\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even}, \\
\{0, \pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd},
\end{cases}
\]

such that \( f^*(v) = \sum_{uv \in E(G)} f(uv) \). We find the values of \( a \) and \( b \) for which the hypercube \( Q_n, n \leq 3 \), is \( (a, b) \)-SEG.