Using a new algorithm (described once in a mail to J. A. Gallian), recently I have completed a verification of the **graceful tree conjecture** for trees with at most n=39 vertices, advancing the recent verification record for n up to 35.

The computation of n=36,37 is done on a T7200. For n=38, it was done with the Amazon EC2 cloud computing service. For n=39, the computation was done on a T7200 then on an i7-3720QM.

Due to heterogeneous computing environment, exact timing is difficult to obtain. A rough estimation is that, the case n=39 took roughly 700 day*core on an i7-3720QM.

An article summarizing my recent verification efforts on graceful tree conjecture and harmonious tree conjecture (in process on yoyo@home) is being prepared.

It would be difficult (at least for me with limited computation resource) to further advance the verification on graceful tree, since the number of trees to verify explodes exponentially. New theorectical results are needed to provide sufficient optimization for further verification to be feasible.