Another equivalent of the graceful tree conjecture (Q2761056)

The well-known graceful tree conjecture says that the vertices of every tree \(T\) of order \(n\) can be labeled \(0,1,\ldots ,n-1\) so that, if \(l\) is the labeling, \(\{|l(y)-l(x)|: xy\in E(T)\}=\{1,2,\ldots ,n-1\}\). The authors set forth a stronger conjecture: every tree \(T\) of order \(n\) with a perfect matching \(M\) has a graceful labeling \(l\) such that, in addition, \(l(x)+l(y)=n-1\) whenever \(xy\in M\). It is proved that both conjectures are equivalent. Some transformations that might be useful in a possible proof of the second conjecture are considered and some families of strongly graceful trees are presented.