On the gracesize of trees

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.