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{\bf Timothy J. Hetherington and Douglas R. Woodall }
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{\bf Edge and Total Choosability of Near-Outerplanar Graphs}
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It is proved that, if $G$ is a $K_4$-minor-free graph with maximum
degree $\Delta \ge 4$, then $G$ is totally $(\Delta+1)$-choosable;
that is, if every element (vertex or edge) of $G$ is assigned a list
of $\Delta+1$ 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. The List-Edge-Colouring Conjecture
is also known to be true for these graphs. As a fairly
straightforward consequence, it is proved that both conjectures hold
also for all $K_{2,3}$-minor free graphs and all $(\bar K_2 + (K_1
\cup K_2))$-minor-free graphs.
\bye