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 \(w \geq 1\). 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.