# Some totally 4-choosable multigraphs

Discussiones Mathematicae Graph Theory (2007)

- Volume: 27, Issue: 3, page 425-455
- ISSN: 2083-5892

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topDouglas R. Woodall. "Some totally 4-choosable multigraphs." Discussiones Mathematicae Graph Theory 27.3 (2007): 425-455. <http://eudml.org/doc/270317>.

@article{DouglasR2007,

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 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.},

author = {Douglas R. Woodall},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {maximum average degree; planar graph; total choosability; list total colouring},

language = {eng},

number = {3},

pages = {425-455},

title = {Some totally 4-choosable multigraphs},

url = {http://eudml.org/doc/270317},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Douglas R. Woodall

TI - Some totally 4-choosable multigraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2007

VL - 27

IS - 3

SP - 425

EP - 455

AB - 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.

LA - eng

KW - maximum average degree; planar graph; total choosability; list total colouring

UR - http://eudml.org/doc/270317

ER -

## References

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