Discussiones Mathematicae Graph Theory 27(3) (2007) 425-455
doi: 10.7151/dmgt.1372

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SOME TOTALLY 4-CHOOSABLE MULTIGRAPHS

Douglas R. Woodall

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

Abstract

It is proved that if G is multigraph with maximum degree 3, and every submultigraph of G has average degree at most 2[1/2] and is different
from one forbidden configuration C4+ with average degree exactly 2[1/2], then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that no two adjacent or incident elements are coloured with the same colour. This shows that the List-Total-Colouring Conjecture, that ch′′(G) = χ ′′(G) for every multigraph G, is true for all multigraphs of this type. As a consequence, if G is a graph with maximum degree 3 and girth at least 10 that can be embedded in the plane, projective plane, torus or Klein bottle, then ch′′(G) = χ′′(G) = 4. Some further total choosability results are discussed for planar graphs with sufficiently large maximum degree and girth.

Keywords: maximum average degree, planar graph, total choosability, list total colouring.

2000 Mathematics Subject Classification: Primary: 05C15; Secondary: 05C35.

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Received 21 April 2006
Revised 6 July 2007
Accepted 6 July 2007