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 Xp(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 X(G) <= Xp(G) <=|V(G)|− a(G)+1, where X(G) and a(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 ceil(log2(2+diam(T))) <= Xp(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.
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- Accepted or Published Version
- DOI:
- Digital Object Identifier (open in new tab) 10.7151/dmgt.1537
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- Category:
- Articles
- Type:
- artykuły w czasopismach recenzowanych i innych wydawnictwach ciągłych
- Published in:
-
Discussiones Mathematicae Graph Theory
no. 31,
pages 183 - 195,
ISSN: 1234-3099 - Language:
- English
- Publication year:
- 2011
- Bibliographic description:
- Borowiecki P., Budajova K., Jendrol S., Krajci S.: Parity vertex colouring of graphs// Discussiones Mathematicae Graph Theory. -Vol. 31., iss. Iss 1 (2011), s.183-195
- DOI:
- Digital Object Identifier (open in new tab) 10.7151/dmgt.1537
- Verified by:
- Gdańsk University of Technology
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