3 problems

A three-distant tree is a tree whose vertices all have distance at most $3$ from a longest path. Gracefulness conjecture for three-distant trees with perfect matchings. Every three…

Let $T$ be a three-distant tree with a longest path $P$. Define the shell of a three-distant tree to be the graph obtained by contracting all branches not on $P$ with size less tha…

Let $T$ be a lobster, and let an almost perfect matching mean a matching that leaves at most one vertex unmatched. Gracefulness conjecture for lobsters with almost perfect matching…