## COLOURING VERTICES OF PLANE GRAPHS UNDER RESTRICTIONS GIVEN BY FACES

Július Czap  and  Stanislav Jendrol'

Institute of Mathematics
P.J. Safárik University
Jesenná 5, SK-04001 Košice, Slovakia
e-mail: julius.czap@upjs.sk
e-mail: stanislav.jendrol@upjs.sk

## Abstract

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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