  This paper deals with two problems about splitting fairly a path with colored
vertices, where "fairly" means that each part contains almost the same amount
of vertices in each color. Our first result states that it is possible to
remove one vertex per color from a path with colored vertices so that the
remaining vertices can be fairly split into two independent sets of the path.
It implies in particular a conjecture of Ron Aharoni and coauthors. The proof
uses the octahedral Tucker lemma. Our second result is the proof of a
particular case of a conjecture of D{\"o}m{\"o}t{\"o}r P{\'a}lv{\"o}lgyi about
fair splittings of necklaces for which one can decide which thieves are
advantaged. The proof is based on a rounding technique introduced by Noga Alon
and coauthors to prove the discrete splitting necklace theorem from the
continuous one.
