The Graceful Tree Conjecture: a class of graceful diameter-6 trees

We survey the current state of progress on the Graceful Tree Conjecture, and then we present several new results toward the conjecture, driven by three new ideas: (1) It has been proven that generalized banana trees are graceful by rearranging the branches at the root–consider rearranging branches at all internal vertices; (2) The method of transfers has typically involved type-1 transfers and type-2 transfers–all type-2 transfers are type-1 transfers in disguise, and hence can be removed from the discussion; (3) The method of transfers has typically used the sequence of transfers $0\rightarrow n\rightarrow 1\rightarrow n - 1\rightarrow\cdots$–transfer backwards to manipulate the resulting labels. Using these ideas, we prove that several classes of diameter-6 trees are graceful, and we generalize some of these classes to larger trees. We also introduce a class of graceful spiders, prove that attaching sufficiently many leaves to any tree gives a graceful tree, and extend known results on trees with perfect matchings.