ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


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


Douglas R. Woodall

School of Mathematical Sciences
University of Nottingham
Nottingham NG7 2RD, UK


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.


[1] N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125-134, doi: 10.1007/BF01204715.
[2] O.V. Borodin, A.V. Kostochka and D.R. Woodall, List edge and list total colourings of multigraphs, J. Combin. Theory (B) 71 (1997) 184-204, doi: 10.1006/jctb.1997.1780.
[3] O.V. Borodin, A.V. Kostochka and D.R. Woodall, Total colourings of planar graphs with large girth, European J. Combin. 19 (1998) 19-24, doi: 10.1006/eujc.1997.0152.
[4] A. Chetwynd, Total colourings of graphs, in: R. Nelson and R.J. Wilson, eds, Graph Colourings (Milton Keynes, 1988), Pitman Res. Notes Math. Ser. 218 (Longman Sci. Tech., Harlow, 1990) 65-77.
[5] P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, in: Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, 1979, Congr. Numer. 26 (1980) 125-157.
[6] M. Juvan, B. Mohar and R. Skrekovski, List total colourings of graphs, Combin. Probab. Comput. 7 (1998) 181-188, doi: 10.1017/S0963548397003210.
[7] L. Lovász, Three short proofs in graph theory, J. Combin. Theory (B) 19 (1975) 269-271, doi: 10.1016/0095-8956(75)90089-1.
[8] V.G. Vizing, Colouring the vertices of a graph in prescribed colours (in Russian), Metody Diskret. Analiz. 29 (1976) 3-10.
[9] W. Wang, Total chromatic number of planar graphs with maximum degree 10, J. Graph Theory 54 (2007) 91-102, doi: 10.1002/jgt.20195.
[10] D.R. Woodall, List colourings of graphs, in: J.W.P. Hirschfeld, ed., Surveys in Combinatorics, 2001, London Math. Soc. Lecture Note Ser. 288 (Cambridge Univ. Press, Cambridge, 2001) 269-301.

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