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On the gracesize of trees

Shoham Letzter, Alexey Pokrovskiy, Ella Williams

TL;DR

This work addresses the asymptotic behavior of the gracesize gs(T) for n-vertex trees by linking graceful labelings to rainbow embeddings in difference-coloured complete graphs. The authors introduce a reduction to an almost-graceful problem, recast it as embedding a tree with many distinct edge colours into a difference-coloured K_{[(1+ε)n]}, and implement a strategy based on splitting the tree into a high-degree core and a small waste set. Central to the approach are tree-splitting lemmas, rainbow matchings in 3-uniform hypergraphs, and a splitting-vertex embedding technique that builds a rainbow T via a rainbow blow-up of an auxiliary forest, followed by careful extension to the full tree. The main contribution is an asymptotic lower bound gs(T) ≥ (1−ε)n for all sufficiently large n, advancing toward the graceful tree conjecture and connecting graceful labellings with extremal and probabilistic embedding methods. This establishes almost-graceful labellings for all large trees and strengthens the bridge between graceful labeling and rainbow subgraph theory.

Abstract

An $n$-vertex tree $T$ is said to be $\textit{graceful}$ if there exists a bijective labelling $φ:V(T)\to \{1,\ldots,n\}$ such that the edge-differences $\{|φ(x)-φ(y)| : xy\in E(T)\}$ are pairwise distinct. The longstanding graceful tree conjecture, posed by Rósa 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 $φ:V(T)\to \{1,\ldots,n\}$. 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.

On the gracesize of trees

TL;DR

This work addresses the asymptotic behavior of the gracesize gs(T) for n-vertex trees by linking graceful labelings to rainbow embeddings in difference-coloured complete graphs. The authors introduce a reduction to an almost-graceful problem, recast it as embedding a tree with many distinct edge colours into a difference-coloured K_{[(1+ε)n]}, and implement a strategy based on splitting the tree into a high-degree core and a small waste set. Central to the approach are tree-splitting lemmas, rainbow matchings in 3-uniform hypergraphs, and a splitting-vertex embedding technique that builds a rainbow T via a rainbow blow-up of an auxiliary forest, followed by careful extension to the full tree. The main contribution is an asymptotic lower bound gs(T) ≥ (1−ε)n for all sufficiently large n, advancing toward the graceful tree conjecture and connecting graceful labellings with extremal and probabilistic embedding methods. This establishes almost-graceful labellings for all large trees and strengthens the bridge between graceful labeling and rainbow subgraph theory.

Abstract

An -vertex tree is said to be if there exists a bijective labelling such that the edge-differences are pairwise distinct. The longstanding graceful tree conjecture, posed by Rósa in the 1960s, asserts that every tree is graceful. The of an -vertex tree , denoted , is the maximum possible number of distinct edge-differences over all bijective labellings . The graceful tree conjecture is therefore equivalent to the statement that for all -vertex trees. We prove an asymptotic version of this conjecture by showing that for every , there exists such that every tree on vertices satisfies . In other words, every sufficiently large tree admits an almost graceful labelling.

Paper Structure

This paper contains 14 sections, 17 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Reduction argument
  3. Rainbow reformulation
  4. Preliminaries
  5. Organisation of the paper
  6. Notation
  7. Probabilistic tools
  8. Proof strategy for \ref{['maintheorem']}
  9. Tree splitting
  10. Finding rainbow subgraphs
  11. Using hypergraphs for matchings
  12. Embedding trees with a splitting vertex
  13. Extending the rainbow embedding
  14. Proof of Lemma 1.7.

Key Result

Theorem 1.2

For all $\varepsilon> 0$ there exist $\eta, n_0 > 0$ such that for all $n>n_0$, the following holds. If $T$ is an $n$-vertex tree and $\Delta(T) \leqslant \frac{\eta n}{\log n}$, then there exists a range-relaxed graceful labelling $\phi : V(T) \rightarrow \{1,\ldots,(1 + \varepsilon)n\}$.

Figures (4)

  • Figure 1: $K_{[5]}$
  • Figure 2: Rainbow copy of $P_5$
  • Figure 3: Rainbow matching between intervals $I_0$ and $I_2$ for $\ell = 7$, with colour set $\{11,12,\ldots,17\}$.
  • Figure 4: Intervals $I_0,\ldots,I_3$ depicted around the semi-circle, with subintervals $I_0'$, $I_2'$ and $I_3'$ highlighted in orange. Vertices in $S$ are embedded into $I_0'$. In this example, $v$ is adjacent to the vertices labelled by $2$ and by $3$ in $T_{\text{aux}}$, and we embed the root vertices corresponding to these vertex sets into $I_2'$ and $I_3'$ respectively. Blue edges have a colour in $C_{02}$, and lighter blue edges have larger colours than darker blue edges. The same holds for pink edges having colours in $C_{03}$.

Theorems & Definitions (44)

  • Conjecture 1.1: Graceful tree conjecture
  • Theorem 1.2: Adamaszek, Allen, Grosu and Hladký adamaszek2020almost
  • Theorem 1.3
  • Lemma 1.4
  • proof : Proof of \ref{['thm:gracesize']} from \ref{['lem:mainGL']}
  • Definition 1.5
  • Conjecture 1.6: Rainbow version of the graceful tree conjecture
  • Lemma 1.6
  • Theorem 2.1: Chernoff bound, see e.g. JansonSvante2000Rg/S
  • Theorem 2.2: McDiarmid's inequality mcdiarmid1989method
  • ...and 34 more