# Splitting necklaces, with constraints

## TL;DR

This paper extends the classical necklace-splitting theorem by introducing new constraints and variants, including equitable, binary, and envy-free splittings, with proofs for prime power and binary cases, and addresses conjectures and preferences.

## Contribution

It presents novel constrained versions of the necklace-splitting theorem, including equitable and envy-free variants, and confirms the binary splitting conjecture for powers of two.

## Key findings

- Existence of fair splittings with approximately equal pieces for prime power number of thieves.
- Confirmation of the binary splitting conjecture for powers of two.
- Envy-free necklace-splitting theorems accommodating individual preferences.

## Abstract

We prove several versions of N. Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace (including "degenerate pieces" if they exist), provided the number of thieves $r=p^\nu$ is a prime power. (2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a $d$-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [7]) in the case $r=2^d$. (3) An interesting variation arises when the thieves have their own individual preferences. We prove several "envy-free fair necklace-splitting theorems" of various level of generality. By specialization we obtain numerous corollaries, among them envy-free versions of (a) "almost equicardinal splitting theorem", (b) "necklace-splitting theorem for $r$-unavoidable preferences", (c) "envy-free binary splitting theorem", etc. As a corollary we also obtain a recent result of Avvakumov and Karasev [1] about envy-free divisions where players may prefer an empty part of the necklace.