Graceful labeling for mushroom trees

Abstract

One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261–290, 1973) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star.