Abstrakt
We show for every k that the binomial tree of order 3k has a vertex-coloring with 2k+1 colors such that every path contains some color odd number of times. This disproves a conjecture of Boroviecki, Budajová, Jenrol and Krajči asserting that for every tree T the minimal number of colors in a such coloring of T is at least the vertex ranking number of T minus one.