On the gracesize of trees

TL;DR

This paper proves that large trees can be almost gracefully labeled, meaning they nearly satisfy the longstanding conjecture that all trees are graceful, by showing their gracesize approaches the maximum possible as the number of vertices grows.

Contribution

The paper establishes an asymptotic version of the graceful tree conjecture, demonstrating that large trees have gracesize close to the number of vertices, which is a significant step forward.

Findings

Every sufficiently large tree has a gracesize at least (1 - ε) times its number of vertices.

The result holds for all ε > 0, with an appropriate threshold n₀ depending on ε.

This provides an asymptotic approximation to the graceful tree conjecture.

Abstract

An $n$-vertex tree $T$ is said to be $\textit{graceful}$ if there exists a bijective labelling $\phi:V(T)\to \{1,\ldots,n\}$ such that the edge-differences $\{|\phi(x)-\phi(y)| : xy\in E(T)\}$ are pairwise distinct. The longstanding graceful tree conjecture, posed by R\'{o}sa in the 1960s, asserts that every tree is graceful. The $\textit{gracesize}$ of an $n$-vertex tree $T$, denoted $\operatorname{gs}(T)$, is the maximum possible number of distinct edge-differences over all bijective labellings $\phi:V(T)\to \{1,\ldots,n\}$. The graceful tree conjecture is therefore equivalent to the statement that $\operatorname{gs}(T)=n-1$ for all $n$-vertex trees. We prove an asymptotic version of this conjecture by showing that for every $\varepsilon>0$, there exists $n_0$ such that every tree on $n>n_0$ vertices satisfies $\operatorname{gs}(T)\geqslant (1-\varepsilon)n$. In other words, every sufficiently large tree admits an almost graceful labelling.