Splitting Necklaces, with Constraints

We prove several versions of Alon's necklace-splitting theorem, subject to additional constraints, as illustrated by the following results. (1) The “almost equicardinal theorem” claims that, without increasing number cuts, one guarantees existence a fair splitting such that each thief is allocated almost same pieces necklace (including “degenerate pieces” if they exist), provided thieves $r=p^\nu$ prime power. By same” we mean for pair them can be given at most piece more (one less) than other. (2) “binary $r=2^d$ and are associated with vertices $d$-cube, then, guarantee adjacent share an edge cube. This result provides positive answer conjecture” in case from Conjecture 2.11 [M. Asada et al., SIAM J. Discrete Math., 32 (2018), pp. 591--610]. (3) An interesting variation arises when have their own individual preferences. envy-free, theorems various level generality, envy-free (a) original (b) (c) binary etc.