Kourovka Notebook Problem 21.130

Definitions and mathematical statement

Let $G$ be a finite abelian group, written additively with identity $0$, and suppose $|G|$ is odd. Let $A\subseteq G$ be a subset of cardinality $n\ge3$. An ordering of $A$ is a list $(a_1,\ldots,a_n)$ containing each member exactly once. Interpret subscripts cyclically by setting $a_{n+1}=a_1$.

The conjecture asks whether there is always an ordering for which \[ a_i+a_{i+1}\ne a_j+a_{j+1} \qquad(1\le i<j\le n). \] Thus all $n$ adjacent sums around the cycle must be distinct, including the closing sum $a_n+a_1$. The sums are computed in $G$, so they use its group operation and any torsion relations of $G$.

Equivalently, consider the complete graph on the elements of $A$, assigning to an edge $\{a,b\}$ the label $a+b\in G$. The required ordering is a Hamilton cycle whose edge labels are all different. This restatement uses undirected edges because $G$ is abelian.

The subset is arbitrary and need not generate $G$, contain $0$, or be a subgroup. Its adjacent sums need not belong to $A$ and may include $0$; only repetition among them is forbidden. The conclusion is existential for each individual subset, and cyclic rotations or reversal of a successful ordering are allowed. Oddness refers to the ambient group order, not to the size $n$ of the subset.

Short English statement

Can every subset of size at least three in a finite abelian group of odd order be cyclically ordered with all neighboring sums distinct?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2639 (KOU-21.130), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.130. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.