On the $Q(a)P(b)$-Super Edge-Graceful $(p,p+1)$-Graphs

Abstract

Let $a$ and $b$ be two positive integers. For the graph $G$ with vertex set $V(G)$ and edge set $E(G)$ with $p=|V(G)|$ and $q=|E(G)|$, we define two sets $Q(a)$ and $P(b)$ as follows:

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

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

For the graph $G$ with $p=|V(G)|$ and $q=|E(G)|$, $G$ is said to be $Q(a)P(b)$-super edge-graceful (in short, $Q(a)P(b)$-SEG), if there exists a function pair $(f,f^{+})$ which assigns integer labels to the vertices and edges; that is, $f^{+}:V(G)\to P(b)$, and $f:E(G)\to Q(a)$ such that $f^{+}$ is onto $P(b)$ and $f$ is onto $Q(a)$, and

$$f^{+}(u)=\sum \{f(u,v):(u,v)\in E(G)\}.$$

We investigate $Q(a)P(b)$ super-edge-graceful labelings for three classes of $(p,p+1)$-graphs.