# Parity vertex colouring of graphs

• Volume: 31, Issue: 1, page 183-195
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Piotr Borowiecki, et al. "Parity vertex colouring of graphs." Discussiones Mathematicae Graph Theory 31.1 (2011): 183-195. <http://eudml.org/doc/271077>.

@article{PiotrBorowiecki2011,
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.},
author = {Piotr Borowiecki, Kristína Budajová, Stanislav Jendrol', Stanislav Krajci},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {parity colouring; graph colouring; vertex ranking; ordered colouring; tree; hypercube; Fibonacci number},
language = {eng},
number = {1},
pages = {183-195},
title = {Parity vertex colouring of graphs},
url = {http://eudml.org/doc/271077},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Piotr Borowiecki
AU - Kristína Budajová
AU - Stanislav Jendrol'
AU - Stanislav Krajci
TI - Parity vertex colouring of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 183
EP - 195
AB - 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.
LA - eng
KW - parity colouring; graph colouring; vertex ranking; ordered colouring; tree; hypercube; Fibonacci number
UR - http://eudml.org/doc/271077
ER -

## References

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8. [8] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes (Morgan Kaufmann, San Mateo, CA, 1992). Zbl0743.68007
9. [9] J.W.H. Liu, The role of elimination trees in sparse factorization, SIAM J. Matrix Anal. Appl. 11 (1990) 134-172, doi: 10.1137/0611010. Zbl0697.65013
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