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Discussiones Mathematicae Graph Theory 31(3) (2011) 441-459
DOI: https://doi.org/10.7151/dmgt.1557

DEFECTIVE CHOOSABILITY OF GRAPHS IN SURFACES

Douglas R. Woodall

School of Mathematical Sciences
University of Nottingham
Nottingham NG7 2RD, UK
e-mail: douglas.woodall@nottingham.ac.uk

Abstract

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ε, and k and d are positive integers such that k ≥ 3 and d is sufficiently large in terms of k and ε, then G is (k,d)^*-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings (choosability). Also, the recent result of Cushing and Kierstead, that every planar graph is (4,1)^*-choosable, is extended to K_3,3-minor-free and K₅-minor-free graphs.

Keywords: list coloring, defective coloring, minor-free graph.

2010 Mathematics Subject Classification: 05C15.

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Received 10 March 2010
Accepted 4 May 2010