Geodesic
Parity vertex colorings of binomial trees
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 177-180

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We show for every k ≥ 1 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 from [1] 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.
Keywords: binomial tree, parity coloring, vertex ranking 
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Gregor, Petr; Škrekovski, Riste. Parity vertex colorings of binomial trees. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 177-180. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a14/