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We prove several versions of N. Alon's "necklace-splitting theorem", subjectto additional constraints, as illustrated by the following results. (1) The"almost equicardinal necklace-splitting theorem" claims that, withoutincreasing the number of cuts, one guarantees the existence of a fair splittingsuch that each thief is allocated (approximately) one and the same number ofpieces of the necklace (including "degenerate pieces" if they exist), providedthe number of thieves $r=p^\nu$ is a prime power. (2) The "binary splittingtheorem" claims that if $r=2^d$ and the thieves are associated with thevertices of a $d$-cube then, without increasing the number of cuts, one canguarantee the existence of a fair splitting such that adjacent pieces areallocated to thieves that share an edge of the cube. This result provides apositive 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 variationarises when the thieves have their own individual preferences. We prove several"envy-free fair necklace-splitting theorems" of various level of generality. Byspecialization we obtain numerous corollaries, among them envy-free versions of(a) "almost equicardinal splitting theorem", (b) "necklace-splitting theoremfor $r$-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 thenecklace.

Splitting necklaces, with constraints