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arXiv · 1308.3038

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

Daniel W. Cranston
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Abstract

The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the set of edges and vertices of $G$ and has an edge between two vertices if their corresponding 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 best previous general upper bound for $\Delta(G) > 3$ was $\floor{\frac32\Delta(G)+2}$, by Borodin et al. When $\Delta(G)=4$, our algorithm gives a better upper bound. When $\Delta(G)\in\{3,5,6\}$, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.).

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Daniel W. Cranston. 2013-08-14. Multigraphs with $\Delta \ge 3$ are Totally-$(2\Delta-1)$-choosable. https://arxiv.org/abs/1308.3038

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