Graceful supertrees with at most $2m-\mathrm{diam}(T)$ edges

Abstract

A graceful labeling of a tree with $m$ edges is a bijection from its vertices to $\{0,1,\ldots,m\}$ whose edge differences are pairwise distinct. The Graceful Tree Conjecture asserts that every tree admits such a labeling; equivalently, the smallest graceful tree containing a given $m$-edge tree has $m$ edges. For this embedding problem, Sethuraman, Ragukumar and Slater proved that fewer than $4m$ edges suffice, while Sethuraman and Ragukumar observed that in practice fewer than $2m$ appear to suffice. We prove the latter bound for every tree. For any root of height $h$ with $t_h$ vertices at maximum distance, we construct a graceful tree containing the original tree with exactly $M=2m-h-t_h+1$ edges; in particular, $2m-\operatorname{diam}(T)$ edges suffice. The construction completes a cyclic reflection of van Bussel's level labeling without increasing its span; each original edge difference $d$ is replaced by $M+1-d$. Unused vertex labels and missing edge differences are indexed by the same intermediate-level positions, allowing an explicit attachment rule that preserves the tree property. The best root for this construction is computable in linear time. Because every graceful supertree labeling restricts to a range-relaxed labeling, a uniform supertree bound of $(2-\varepsilon)m+O(1)$ would imply such a range-relaxed bound for all trees.