A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χₚ(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χₚ(G) ≤ |V(G)|-α(G)+1, where χ(G) and α(G) are the chromatic number and the independence number of G, respectively. The bounds are improved for trees. Namely, if T is a tree with diameter diam(T) and radius rad(T), then ⌈log₂(2+diam(T))⌉ ≤ χₚ(T) ≤ 1+rad(T). Both bounds are tight. The second thread of this paper is devoted to relationships between parity vertex colourings and vertex rankings, i.e. a proper vertex colourings with the property that each path between two vertices of the same colour q contains a vertex of colour greater than q. New results on graphs critical for vertex rankings are also presented.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1537,
author = {Piotr Borowiecki and Krist\'\i na Budajov\'a and Stanislav Jendrol' and Stanislav Krajci},
title = {Parity vertex colouring of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {183-195},
zbl = {1284.05091},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1537}
}
Piotr Borowiecki; Kristína Budajová; Stanislav Jendrol'; Stanislav Krajci. Parity vertex colouring of graphs. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 183-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1537/
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