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On the Graham--Sloane harmonious labelling conjecture

Alp Müyesser, Alexey Pokrovskiy

TL;DR

The paper addresses the problem of when an $n$-vertex tree $T$ can be labelled by a finite abelian group $G$ of order $n$ so that edge-sums are all distinct, reframing this as finding rainbow copies of $T$ in the Cayley-sum graph $K_G$. It introduces a core-based, prescribed-sets embedding theorem that reduces the problem to embedding a small core $T_{ ext{core}}$ with global sum constraints, coupled with probabilistic completion lemmas to extend to the entire tree; this yields a precise obstruction-based characterisation for bounded-degree trees. Consequently, for any fixed maximum degree $ ext{Δ}$ there exists $n_0$ such that every $n$-vertex tree with $ ext{Δ}(T)\le ext{Δ}$ admits a rainbow embedding into $K_G$ when $|G|=n$, thereby confirming Graham–Sloane and Chang–Hsu–Rogers for bounded-degree trees. The results further connect to orthogonal double covers and multiple conjectures in graph labelings and decompositions, offering a robust framework for harmonic labellings and rainbow-embedding problems. Overall, the work provides a sharp, core-driven method to resolve harmonious labelling questions in a broad, structured regime and highlights new obstructions along with their absence as the key dichotomy.

Abstract

Consider an order $n$ abelian group $G$ and a tree $T$ on $n$ vertices. When is it possible to (bijectively) label $V(T)$ by $G$ so that along all edges $xy$ of $T$, the sums $x+y$ are distinct? This problem can be traced back to the work of Graham and Sloane on the harmonious labelling conjecture, and has been studied extensively since its introduction in 1980. We give a precise characterisation that holds for all bounded degree trees. In particular, our characterisation implies that if $G=\mathbb{Z}/n\mathbb{Z}$ and $T$ is a bounded degree tree, the desired labelling exists. This confirms a conjecture of Graham and Sloane from 1980, and another conjecture of Chang, Hsu, and Rogers from 1987, for bounded degree trees. Our results also have further applications for the study of graph coverings.

On the Graham--Sloane harmonious labelling conjecture

TL;DR

The paper addresses the problem of when an -vertex tree can be labelled by a finite abelian group of order so that edge-sums are all distinct, reframing this as finding rainbow copies of in the Cayley-sum graph . It introduces a core-based, prescribed-sets embedding theorem that reduces the problem to embedding a small core with global sum constraints, coupled with probabilistic completion lemmas to extend to the entire tree; this yields a precise obstruction-based characterisation for bounded-degree trees. Consequently, for any fixed maximum degree there exists such that every -vertex tree with admits a rainbow embedding into when , thereby confirming Graham–Sloane and Chang–Hsu–Rogers for bounded-degree trees. The results further connect to orthogonal double covers and multiple conjectures in graph labelings and decompositions, offering a robust framework for harmonic labellings and rainbow-embedding problems. Overall, the work provides a sharp, core-driven method to resolve harmonious labelling questions in a broad, structured regime and highlights new obstructions along with their absence as the key dichotomy.

Abstract

Consider an order abelian group and a tree on vertices. When is it possible to (bijectively) label by so that along all edges of , the sums are distinct? This problem can be traced back to the work of Graham and Sloane on the harmonious labelling conjecture, and has been studied extensively since its introduction in 1980. We give a precise characterisation that holds for all bounded degree trees. In particular, our characterisation implies that if and is a bounded degree tree, the desired labelling exists. This confirms a conjecture of Graham and Sloane from 1980, and another conjecture of Chang, Hsu, and Rogers from 1987, for bounded degree trees. Our results also have further applications for the study of graph coverings.

Paper Structure

This paper contains 13 sections, 35 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. On the conjectures of Graham--Sloane, Chang--Hsu--Rogers, Andersen and Montgomery--Pokrovskiy--Sudakov
  4. Orthogonal double covers
  5. Overview and the core lemma
  6. Preliminaries
  7. Completion lemmas
  8. Embeddings and injections
  9. Approximations of trees
  10. Tree embeddings using prescribed vertices and colours
  11. Characterizing harmonious trees
  12. Concluding remarks
  13. Appendix: approximate tree embeddings

Key Result

Theorem 1.5

For any $\Delta$, there exists a $n_0$ sufficiently large so that the following holds for any $n\geq n_0$. Let $T$ be an $n$-vertex tree with $\Delta(T)\leq \Delta$ and $G$ an abelian group of size $n$. There is a rainbow copy of $T$ in $K_G$ if, and only if, we have none of the following:

Theorems & Definitions (93)

  • Conjecture 1.1: The harmonious labelling conjecture, graham1980additive
  • Definition 1.2
  • Conjecture 1.3
  • Example 1.4
  • proof
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Example 1.8
  • proof
  • ...and 83 more