Some a-Graphs and Odd Graceful Graphs

Abstract

We show that if $G$ has an odd graceful labeling $f$ such that $max\{f(x):f(x)\text{ is even},x\in A\}<min\{f(x):f(x)\text{ is odd},x\in B\}$, then $G$ is an o-graph, and if $G$ is an a-graph, then $G\odot K_n$ is odd graceful for all $w\ge 1$. Also, we show that if $G_1$ is an a-graph and $G_2$ is an odd graceful, then $G_1\cup G_2$ is odd graceful. Finally, we show that some families of graphs are a-graphs and odd graceful.