Total 4-choosability of series-parallel graphs
The electronic journal of combinatorics, Tome 13 (2006)
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It is proved that, if $G$ is a $K_4$-minor-free graph with maximum degree 3, 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 every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ${\rm ch}"(G) = \chi"(G)$ for every graph $G$, is true for all $K_4$-minor-free graphs and, therefore, for all outerplanar graphs.
Douglas R. Woodall. Total 4-choosability of series-parallel graphs. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1123
@article{10_37236_1123,
author = {Douglas R. Woodall},
title = {Total 4-choosability of series-parallel graphs},
journal = {The electronic journal of combinatorics},
year = {2006},
volume = {13},
doi = {10.37236/1123},
zbl = {1113.05040},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1123/}
}
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