ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 31(1) (2011) 183-195
DOI: 10.7151/dmgt.1537


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

Kristína Budajová

Faculty of Aeronautics
Technical University Košice
Rampová 7, SK-04121 Košice, Slovak Republic

Stanislav Jendrol'

Institute of Mathematics
P.J. Safárik University Košice
Jesenná 5, SK-04154 Košice, Slovak Republic

Stanislav Krajci

Institute of Computer Sciences
P.J. Safárik University Košice
Jesenná 5, SK-041 54 Košice, Slovak Republic


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.


[1] H.L. Bodlaender, J.S. Degoun, K. Jansen, T. Kloks, D. Kratsch, H. Müller and Zs. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168-181, doi: 10.1137/S0895480195282550.
[2] D.P. Bunde, K. Milans, D.B. West and H. Wu, Parity and strong parity edge-colorings of graphs, Congr. Numer. 187 (2007) 193-213.
[3] D. Dereniowski, Rank colouring of graphs, in: M. Kubale ed., Graph Colorings, Contemporary Mathematics 352 (American Mathematical Society, 2004) 79-93.
[4] R. Diestel, Graph Theory (Springer-Verlag New York, Inc., 1997).
[5] G. Even, Z. Lotker, D. Ron and S. Smorodinsky, Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks, SIAM Journal on Computing 33 (2003) 94-136, doi: 10.1137/S0097539702431840.
[6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980).
[7] M. Katchalski, W. McCuaig and S. Seager, Ordered colourings, Discrete Math. 142 (1995) 141-154, doi: 10.1016/0012-365X(93)E0216-Q.
[8] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes (Morgan Kaufmann, San Mateo, CA, 1992).
[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.
[10] A. Sen, H. Deng and S. Guha, On a graph partition problem with application to VLSI layout, Inform. Process. Lett. 43 (1992) 87-94, doi: 10.1016/0020-0190(92)90017-P.

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