ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2023): 0.5

5-year Journal Impact Factor (2023): 0.6

CiteScore (2023): 2.2

SNIP (2023): 0.681

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 29(3) (2009) 521-543


Július Czap  and  Stanislav Jendrol'

Institute of Mathematics
P.J. Safárik University
Jesenná 5, SK-04001 Košice, Slovakia


We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions.

Keywords: vertex colouring, plane graph, weak parity vertex colouring, strong parity vertex colouring, proper colouring, Lebesgue theorem.

2000 Mathematics Subject Classification: 05C10, 05C15.


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Received 28 March 2008
Revised 19 August 2008
Accepted 5 September 2008