Graceful Tree Conjecture is one of my favorite open problems (See this earlier post). Trees with diameter 4 and 5 are known to be graceful a decade ago.
One of my advisees, Matt Superdock, made progress towards proving that all diameter 6 trees are graceful. Matt is an undergraduate senior in our Mathematics Department. He proved that an interesting class of diameter 6 trees are graceful. His thesis is available here.
I hope his work motivates more researchers to make progress towards resolving Graceful Tree Conjecture, particularly for diameter 6 trees. Matt’s work really tests the limit of current techniques. Perhaps we need new techniques/insights to prove that all diameter 6 trees are graceful.
My Brain is Open 