k-graceful labeling for the ladder graph and the roach graph

A simple graph is said to be a vertex -graceful if there exists a vertex graceful labeling on the vertices of . A graceful labeling of is a vertex labeling be an injective mapping from to such that the edge labeling defined by is also injective. When , is called graceful labeling, and is called a graceful graph. There is a very famous conjecture in this area that every tree is graceful. Numerous studies have been conducted in this area over the past few decades, and several results have been obtained. In this research work, we prove that the ladder graph admits the -graceful labeling. The ladder graph is a graph obtained from the Cartesian product of and Moreover, we studied the -gracefulness of the roach graphs and could obtain some partial results. However, we strongly believe that every roach graph is -graceful. Finally, we introduced two open problems for future works