We study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno's formula for the Ganter-Kapranov 2-character of a finite group.

warwick.ac.uk/lib-publicationsOriginal citation: Wendland, Alex. (2015) Colouring of plane graphs with unique maximal colours on faces. Journal of Graph Theory. Permanent WRAP URL:http://wrap.warwick.ac.uk/73823 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher's statement:"This is the peer reviewed version of the following article: Wendland, Alex. (2015) Colouring of plane graphs with unique maximal colours on faces. Journal of Graph Theory., which has been published in final form http://dx.doi.org/10.1002/jgt.22002 . This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for SelfArchiving." A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.uk Colouring of plane graphs with unique maximal colours on facesAlex Wendland * AbstractThe Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colours. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1, . . . , 4 in such a way that every face contains a unique vertex coloured with the maximal colour appearing on that face. They proved that every plane graph has such a colouring with the numbers 1, . . . , 6. We prove that every plane graph has such a colouring with the numbers 1, . . . , 5 and we also prove the list variant of the statement for lists of sizes seven.