Fair splitting of colored paths

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 Dömötör Pálvölgyi 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.