We prove several versions of 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 almost 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. By “almost the same” we mean that for each pair of thieves one of them can be given at most one piece more (one piece less) than the other. (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” in the case $r=2^d$ from Conjecture 2.11 in [M. Asada et al., SIAM J. Discrete Math., 32 (2018), pp. 591--610]. (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, as illustrated by the envy-free versions of (a) Alon's original necklace-splitting theorem, (b) the almost equicardinal splitting theorem, and (c) the binary splitting theorem, etc.

A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, to a flat position, in such a way that the paper is not torn, stretched and, for rigid origami, not bent anywhere except along the given creases.