## PARITY VERTEX COLOURING OF GRAPHS

 Piotr Borowiecki Department of Algorithms and System Modeling Faculty of Electronics, Telecommunications and Informatics Gdask University of Technology Narutowicza 11/12, 80-233 Gdask, Poland e-mail: pborowie@eti.pg.gda.pl Kristína Budajová Faculty of Aeronautics Technical University Košice Rampová 7, SK-04121 Košice, Slovak Republic e-mail: kristina.budajova@tuke.sk Stanislav Jendrol' Institute of Mathematics P.J. Safárik University Košice Jesenná 5, SK-04154 Košice, Slovak Republic e-mail: stanislav.jendrol@upjs.sk Stanislav Krajci Institute of Computer Sciences P.J. Safárik University Košice Jesenná 5, SK-041 54 Košice, Slovak Republic e-mail: stanislav.krajci@upjs.sk

## Abstract

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.

Keywords: parity colouring, graph colouring, vertex ranking, ordered colouring, tree, hypercube, Fibonacci number.

2010 Mathematics Subject Classification: 05C15, 05C05, 05C90, 68R10.

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Received 1 December 2009
Revised 12 May 2010
Accepted 12 May 2010