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 C⁺₄ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1372,
author = {Douglas R. Woodall},
title = {Some totally 4-choosable multigraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {27},
year = {2007},
pages = {425-455},
zbl = {1142.05031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1372}
}
Douglas R. Woodall. Some totally 4-choosable multigraphs. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 425-455. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1372/
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