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The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the setof edges and vertices of $G$ and has an edge between two vertices if theircorresponding elements are either adjacent or incident in $G$. We show that if$G$ has maximum degree $\Delta(G)$, then $T(G)$ is $(2\Delta(G)-1)$-choosable.We give a linear-time algorithm that produces such a coloring. The bestprevious general upper bound for $\Delta(G) > 3$ was$\floor{\frac32\Delta(G)+2}$, by Borodin et al. When $\Delta(G)=4$, ouralgorithm gives a better upper bound. When $\Delta(G)\in\{3,5,6\}$, ouralgorithm matches the best known bound. However, because our algorithm issignificantly simpler, it runs in linear time (unlike the algorithm of Borodinet al.).

Multigraphs with $\Delta \ge 3$ are Totally-$(2\Delta-1)$-choosable