SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1268-1286, January 2021.
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.

中文翻译：

---

分裂项链，有约束

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1268-1286 页，2021 年 1 月。
我们证明了 Alon 的项链分裂定理的几个版本，受到额外的约束，如下面的结果所示。(1)“几乎等心数项链分裂定理”声称，在不增加切割次数的情况下，可以保证公平分裂的存在，使得每个小偷分配到几乎相同数量的项链（包括“退化碎片”） ”如果它们存在），前提是盗贼的数量 $r=p^\nu$ 是素数。“几乎相同”的意思是，对于每一对盗贼，他们中的一个最多可以比另一个多一件（少一件）。(2) “二元分裂定理”声称，如果 $r=2^d$ 并且小偷与 $d$-cube 的顶点相关联，那么，在不增加切割次数的情况下，人们可以保证公平分割的存在，这样相邻的碎片被分配给共享立方体边缘的小偷。该结果为 [M. Asada 等人，SIAM J. Discrete Math., 32 (2018), pp. 591--610]。(3) 当窃贼有自己的个人喜好时，就会出现一个有趣的变化。我们证明了几个具有不同普遍性的无嫉妒、公平的项链分裂定理，如 (a) 阿隆原始项链分裂定理的无嫉妒版本、(b) 几乎等心数分裂定理和 (c)二元分裂定理等。该结果为 [M. Asada 等人，SIAM J. Discrete Math., 32 (2018), pp. 591--610]。(3) 当窃贼有自己的个人喜好时，就会出现一个有趣的变化。我们证明了几个具有不同普遍性的无嫉妒、公平的项链分裂定理，如 (a) 阿隆原始项链分裂定理的无嫉妒版本、(b) 几乎等心数分裂定理和 (c)二元分裂定理等。该结果为 [M. Asada 等人，SIAM J. Discrete Math., 32 (2018), pp. 591--610]。(3) 当窃贼有自己的个人喜好时，就会出现一个有趣的变化。我们证明了几个具有不同普遍性的无嫉妒、公平的项链分裂定理，如 (a) 阿隆原始项链分裂定理的无嫉妒版本、(b) 几乎等心数分裂定理和 (c)二元分裂定理等。