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 $f^{\ast }:V(G)\to P(b)$, where

$$Q(a)=\{\begin{array}{ll}\{\pm a,\pm (a+1),\dots ,\pm (a+(q-2)/2)\}& \text{if }q\text{ is even},\\ \{0,\pm a,\pm (a+1),\dots ,\pm (a+(q-3)/2)\}& \text{if }q\text{ is odd},\end{array}$$

$$P(b)=\{\begin{array}{ll}\{\pm b,\pm (b+1),\dots ,\pm (b+(p-2)/2)\}& \text{if }p\text{ is even},\\ \{0,\pm b,\pm (b+1),\dots ,\pm (b+(p-3)/2)\}& \text{if }p\text{ is odd},\end{array}$$

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