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Discussiones Mathematicae Graph Theory 32(1) (2012)
177-180
DOI: https://doi.org/10.7151/dmgt.1595
Parity Vertex Colorings of Binomial Trees
| Petr Gregor
Department of Theoretical Computer Science and Math. Logic | Riste Škrekovski
Department of Mathematics |
Abstract
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
2010 Mathematics Subject Classification: 05C15, 05C05, 05C90, 68R10.
References
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| [2] | D.P. Bunde, K. Milans, D.B. West and H. Wu, Parity and strong parity edge-colorings of graphs, Combinatorica 28 (2008) 625--632, doi: 10.1007/s00493-008-2364-3. |
| [3] | A.A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211--249, doi: 10.1023/A:1010767517079. |
Received 25 October 2010
Revised 10 February 2011
Accepted 10 February 2011
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