五个星之并的优美性

The Gracefulness of the Union of Five Stars

在解决了二个星之并、三个星之并（二个星之并、三个星之并不全是优美图）四个星之并的优美性问题之后，证明了任意五个星之并都是优美图，因其优美标号的构造很复杂，故分成多种情况逐一加以解决。

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.