The Gracefulness of the Union of Five Stars

The graceful graph is a subject with wide-ranging applications. It has developed quickly in recent years. In this paper we solved the gracefulness of the union of two staro, the union of three stars (the union of two stars and the union of the three stars are not always graceful however) and the union of four stars. However, no matter how the stars unite it has been proved that the union of five stars creates only graceful graphs. Since they are gracefol, labellings are complicated. Thus, we divide them into many cases first, and then, solve them one by one.